"Traditional and stylistic reasons", as un-quantifiable and non-scientific as they may be, are nevertheless a large part of it. While I enjoy H&R single shot rifles, one of their primary attractions is their low price...it encourages experimentation, since little is lost if an experiment fails. Ruger #1's cost 3 or 4 times as much, and yet many folks, I among them, buy them just because we like them. If your purchase of a single shot rifle is based upon its style, its classic appearance, or anything else other than raw performance, then the odds are overwhelmingly in favour of your choosing a similarly "classic" cartridge to complete the package. It doesn't matter if it doesn't make strict logical sense, and frankly it doesn't matter if you don't want to believe it...the simple fact of what sells and what doesn't speaks for itself. There's a reason why Ruger has not, and will not in future, chamber the #1 in .300AAC: because virtually no-one will buy it.
But forget that aspect. Wasn't the .300AAC designed to fit into an AR action with minimal modifications, and/or to efficiently produce decent sub-sonic performance? I seem to recall reading that somewhere, and frankly I don't care enough to bother checking it now. But it's design criteria certainly differ vastly from the design criteria of a single-shot rifle with turn-of-the-19th-century styling. You compare it to the .30-30, as though that were a fantastic hunting cartridge, but the .30-30 is really only popular because of the lever-action rifles that chamber it. It's adequate for short-to-medium range and medium game, but it would be almost as functionally useless in the #1 as the AAC...but would probably at least sell, because its vintage and long pedigree might appeal to some lovers of old-time cartridges.
As long as you want to get something that is so unique as to be almost ludicrous, why not go the other way? Build a nice AR chambered in, say, 7x57 or maybe .218Bee. Sure, you can do it, but should you do it? You'll have essentially the same thing, i.e. a gun that you won't be able to give away. And again, there's nothing wrong with that, as long as you intend to keep it forever, or don't mind throwing more good money after bad to convert it to something more saleable down the road. Either way, good luck with your choice...and good luck looking for someone to agree with you, aside from anonymous internet personae who will urge you on to do anything at all, simply because they have no horse in the race.