I agree. I get the sense that some Golden Age creators scripted with an eye towards inspiring imaginative play that their target audience engaged in. I perceived this in The Spectre stories and I sense it here. It may just be that I myself played superhero when I was little and so I'm attributing conscious intent where it really wasn't, but when I see stories incorporating magic pills, chemistry sets, radio messages, puppets, gimmicked rings, I think about how many kids, especially in an era when passive entertainment like videos and computer games didn't consume kids' recreation time like it does today, would have at hand the means of playing out the scenarios they read about in their comics--and would be more inclined to actually do so.

I suspect Roy's primary motivation was explaining why Hour-Man was unceremoniously dropped from the JSA. While the source material certainly didn't suggest any addiction, the pieces of the puzzle do fall together neatly, considering that it is immediately after his departure from the JSA that he adopts an alternative ray-based means of gaining powers. "See, I'm not taking pills any more! I'm not addicted!" And as we know, Rex's timid nature is being phased out, which could imply, as I've suggested, that Miraclo is having a sustained effect on his personality, or alternatively, that he's on Miraclo more often than we are shown (remember, the one-a-day limit was also a late addition to the premise). 
I've done some thinking on Hour-Man's JSA membership. Why would the team want someone whose powers can expire at inconvenient times? I'll be sharing my thinking on that when we get to ALL STAR COMICS #3...

I'm not arguing that it worked; the Minute Men might have been an effective component to a more appealing feature than Hour-Man, but it doesn't seem to have been a successful angle for this strip, and they don't appear to have really tried. As depicted, the Minute Men is essentially a local kid's gang, not a nation-wide army, and Hour-Man doesn't venture outside his home base, which would have given the concept a little more punch. But it's hard to disagree with your thesis: Rex is not someone most kids would have wanted to be sidekicks to. 
It seems like there must be more superheroes with time-limitations, but the fact that it's hard to come up with others than Underdog and Mr. Terrific suggests that they indeed weren't hits. 
I watched Mr. Terrific as a kid, but I never since noticed that the character nicked from two DC/AA Golden Agers, using the name of one and the gimmick of the other. 

Green Lantern has always had a time limit.... Hal charging his ring once a day was great for story telling, it could be time to charge at any dramatic moment, (and often was in the first several stories). Even later when they got off the once every 24 hours thing, the rings still run out of juice. The newest variation in Sojourner (from Far Sector) is that the ring recharges slowly over time (no battery).

If you want to get away from DC... Popeye wasn't spinach powered all the time.. just for a bit. Tony Stark in the early days ran out of power pretty regularly, more often when they were still doing the thing were the suit was keeping sharpnel away from his heart with 'magnetism'.

Didn't Hulk only come out at night at first? Seems familiar. Then of course Vampires... definitely plenty of examples!

ADVENTURE COMICS #54 September 1940:



And the introduction gets even more long-winded: 'Presenting "Tick-Tock Tyler", (The Man of the Hour) as Hour-Man with Minute-Man Martin by Bernard Baily'



by Ken Fitch and Bernard Baily

SYNOPSIS:

The story opens in the home of "Banker Morvan" (way to get the exposition over with, "Writer Fitch"!), where the banker and his wife are ordering their over-privileged son Rodney to resign from the new Minute-Men of America ("And this Hour-Man! Bosh, a grown man with such ideas!"). Rodney argues that they're fine fellows, "especially Captain Martin!"

Jimmy Martin is disappointed that Rodney won't be coming to the meeting. He refuses to accept the return of Rodney's portable radio (so Hour-Man has shelled out to equip these thousands of boys with their own radios? Quite generous of him!) and Jimmy promises to ask Hour-Man to intervene on Rodney's behalf. (So this club has a "national scale" according to the introductory caption, but it's apparently particularly active in Hour-Man's home town.)

When Jimmy heads off, Rodney is kidnapped for ransom, the latest in an ongoing string, as Rex's unnamed Boss is noting back at the lab. That evening, Hour-Man "waits for Jimmy" (I guess he does one-on-one confabs with "Captain" Martin), and doesn't go to the club meeting, which must have been the previous day, because Jimmy brings with him a newspaper announcing Rodney's kidnapping. Most alarming is that Banker Morvan is refusing to pay the ransom for the recently-resigned Minute Man.

Hour-Man pops a Miraclo and heads over to Moran's to convince him to pay up. We see a little chemistry in action when Rex knocks out a guard, with sleeping gas instead of tear gas in his ring this time. Rex abducts Morvan, leaping (with Morvan in his arms) via a pool's diving board onto a rooftop to elude Morvan's pursuing security detail.



Upon interrogation, Morvan explains to Rex that he's not paying the ransom "because I'm a citizen and the police should protect me!" Rex still doesn't quite get it, but strands Morvan on the roof, departing with a promise to find Rodney. Doing detective work at the schoolyard, Rex finds "crescent shaped tire marks", and calls on his Minute Men (via his portable radio) to be his primitive search engine, enquiring en masse at all the tire stores in town.

The Minute Men get to business, and one of them (a currently unnamed fellow in a checkered hat with a turtleneck pulled up over his mouth) hits paydirt, learning that such tires were sold to Ricky Mafirri owner of the Blue Goose Nite Club. Meanwhile, Maffiri has had Morvan himself kidnapped (I guess he got down off that roof), and Morvan is tearfully reunited with his son.

Locked up together, Rodney finally remembers he still has the portable radio with which to contact Hour-Man, but their captors catch him and confiscate the transmitter. Hour-Man has arrived at the Blue Goose, where he suspects Rodney's being held, but his Miraclo inconveniently wears off, and darned if he didn't forget to bring his meds with him!



Powerless and confined with the Morvans, Hour-Man uses the miniature radio in his costume to contact Jimmy and arrange for Jimmy to retrieve the pills and deliver them through a small window in the club. Jimmy arrives with not just the pills, but a coterie of Minute Men (including the short turtleneck kid).



With a reinvigorated Hour-Man, they all give the criminals a sound thrashing, except for Maffiri, who flees by car (of course!). As we know by now, if Hour-Man excels at anything, it's racing after speeding cars; not only does he catch up with the kidnapper, he shoves it crashing directly into the police station!

Rodney's grateful parents relent and allow their son to rejoin the Minute Men. The story closes with an offer to the readers: one dollar and an autographed picture of Hour-Man and Minute Man Martin for the ten best letters on "how you like the Minute Men of America"!

COMMENTARY:

For the moment, Fitch is pushing the Minute Men hard as the new focus of the strip. I'm obliged to note the flimsy foundations of this development, foremost how Hour-Man has come to be the hero of thousands of boys across the nation with only a handful of adventures, most of them unremarkable, one of them resulting in bad publicity, and all of them localized. Superman would certainly be making the news in every paper across the country, Hour-Man's lucky to get headlines in his home town of Appleton (although the papers there do seem to be hard up for news!). As I implied before, equipping his fans with radios would be expensive, but maybe only the local boys got those perks. Notice also that Rex doesn't know the members individually, except for "Captain Martin". I think it would lead to some disgruntled ranks for Martin to have such a privileged relationship with the leader. (If I were Jimmy, I'd have been tempted to nick one or two of those magic pills for myself!) That kind of favoritism is already watering down the idea of an army of sidekicks in favor of the tired trope of the single kid sidekick, in only the second installment. Jimmy better not get used to being Hour-Man's Number Two, because the weird kid wearing his turtleneck over his mouth is gunning for the gig.

Speaking of which, was that ever a thing? Wearing a turtleneck rolled up over your lower face? I remember one of the guys in "Fat Albert and the Cosby Kids" sported the same affectation.

While Fitch is no longer depicting Rex as a "coward when unpowered", he's still putting some curiously unheroic ideas into his hero's outlook. Like the first instinct being to pay off kidnappers!? And rather than pursue the kidnappers, Hour-Man instead abducts and terrorizes the victim's dad!? Presumably the previous kidnappings had all been satisfactorily resolved with pay-offs, so Hour-Man never bothered to intervene in those cases.

So here's the second story in which unique tire treads have been a clue. This time, it gives an excuse to send out the radio call. I can't help but imagine the poor employees at the tire stores running off the thirteenth brat to come in asking about crescent-shaped tread sales. And wait a second, even if such treads were indeed "unusual", tires retain their treads for a very long time. There's no reason given to discard the likelihood that these tires were purchased years ago! I guess I'll do Ken Fitch's job and add in the detail that Hour-Man recognized these as an expensive variety of tire just introduced into the market this week.

Miraclo wears off again, emphasizing what a poor planner Rex Tyler is. It's hard to believe that Rex wouldn't be keeping a pocketful on him, at least when he's suited up. It's also hard to believe that Banker Morvan, even with feelings of gratitude, would be happy to have his son join a club that engaged in activities like beating the tar out of criminals with their baseball bats.

While I'm enjoying Baily's art even at this primitive point in his development, I'll note that he doesn't seem concerned about depicting these adventures in a consistent environment. The Morvan estate is on a vast plain with just a few trees in the distance, and judging from the assault from the Minute Men, the villain's nightclub was nestled on the side of a rocky, snow-peaked mountain range!