Wouldn't that exact same problem you mention happen with that "only new Pokémon before the first gym" approach?
I guess it would result in a ninth bird, yes. That's a downside.
But on the upside, it means the developers couldn't be content with introducing
just the bird, bug and rodent plus a couple of crapmons and then pad out the encounters with Pokémon from the earlier games. Sun and Moon in particular fall hard into this trap. On Route 1 your Alolan options are limited to just Yungoos, Pikipek, and Grubbin, plus Rattata-A if you're willing to count that as a new Pokémon. Contrast all the old-gen Pokémon you meet there: Caterpie (and Metapod), Ledyba, Spinarak, Pichu (and Pikachu by SOS call), Bonsly (and Sudowoodo by SOS call), Happiny, Munchlax (and Snorlax by SOS call), Slowpoke, and Wingull (that's 13 in case you didn't count). USUM adds Buneary and Inkay to the route too. Moving on to the Trainer's School, you can find the Galarian forms of Meowth and Grimer, plus Magnemite. Zorua too in USUM. In Hau'oli City, the Pokémon added to the selection are Abra, Mime Jr./Mr. Mime (USUM), and Furfrou (USUM). Onto Route 2: Drowzee, Smeargle, Spearow, Growlithe, Makuhita, Machop (Trade), Ekans (USUM), Dunsparce (USUM), and Hawlucha (USUM, Trade). Meanwhile only Cutiefly and Crabrawler are added as Alolan Pokémon (and you're not guaranteed to find the latter, as it's only found in berry piles). You might want to stop by Hau'oli Cemetery too, where the selection is: Zubat, Gastly, Misdreavus, Drifloon, and Murkrow (USUM). That's all the locations up until the first Trial.
Two towns and two routes into the game, and you've only encountered five Gen VII Pokémon, three of which are the boring ol' bird, bug, and rodent. Plus three Kanto Pokémon dipped in paint, if you count those as new. Meanwhile, you've seen up to 35 old-gen Pokémon. Mind you, this is the hellish slogscape of Gen VII, so you've taken a fair while to get here as well. Granted, this does set the tone for the rest of Alola. The new Gen VII Pokémon are nowhere to be seen, and the old-gen Pokémon offer much greater versatility anyway. The only remaining new Gen VII Pokémon to be found on all of Melemele are Oricorio and Rockruff (the latter only after you get Tauros). Kalos is actually even worse, with only Scatterbug, Fletchling, Bunnelby, and Litleo being Gen VI options before the first Gym.
To be fair to SwSh for once, I think they struck a quite nice balance here. On Route 1 and 2 you find Blipbug, Rookidee, Skwovet, Wooloo, Nickit, Chewtle, and Yamper, plus Arrokuda if you're willing to fish for a while. There's Galarian Zigzagoon too, but it's rare. The Gen VIII 'mons outnumber the earlier-game 'mons (which are Caterpie, Hoothoot, Grubbin, Lotad/Seedot, Purrloin, and Magikarp) in the very beginning, and only about half of them are archetypes (coincidentally, so is half the old-gen Pokémon). The Pokémon found on the first two routes aren't top shelf picks from old generations either, so it's not like you'd be much better off with them over the native Galarian Pokémon. And then the Wild Area opens up and you're getting access to potentially hundreds of Pokémon before the first Gym. Still, I think they would have benefited from having a couple more native Pokémon to offer at this earliest point of the game, especially in the Wild Area. Looking through the encounter lists for the Wild Area, as far as I can tell, when you first get to the area you can't find a single new Gen VIII Pokémon apart from ones you already saw on the first two Routes, possibly except in Raids. Galar should have had a few more natives to entice you to search the Wild Area when you first get there.
For the old-school linear games, let's appreciate RSE for a bit here. The first Route is rather boring, but the regional bug and rodent have Poochyena for company to break up the monotony. On Route 102, Lotad and Seedot make their appearance (three-stage evolutions! Nice!), alongside Ralts (even better!) and Surskit if you're
really willing to search for it (1% encounter, woo!). Route 103 and 104 give you Wingull and Taillow too. Plus Marill, for whatever reason, although only in Emerald. Petalburg Woods lets you pick up Shroomish and possibly Slakoth if you're patient (and you better be if you plan to use Slakoth in the first place ...). Before the first Gym you can also explore Route 116 for a chance to find Nincada, Whismur, Skitty, and, in Emerald, Abra. That's 14 new evolution families (and two old ones in Emerald) before the first Gym. Yes, there are regional bugs, birds, and rodents among them (arguably two regional birds, even), but also ten other families - Lotad and Seedot being version exclusives.
The set-up used in Hoenn requires a large roster of new Pokémon, to give you plenty of options early. It's another benefit of having a large generational Pokédex, you get to have room for that sort of thing. Of course, Unova managed to somehow bungle this up with the exact same premise (exhaustive list of Pokémon available before the first Gym: Starter, Patrat, Lillipup, Audino, Purrloin, Pidove, Elemental Monkey) despite its ginormous dex size, so it's not foolproof.
Gotta give Gen I a quick mention too. It lets you have access to Rattata, Pidgey, Nidoran-M, Nidoran-F, Mankey, Spearow, and Caterpie/Weedle before the first Gym. For the very beginning of the franchise, it's not half bad. The bugs, birds, and rodents overwhelm the first routes, but that little detour into Route 22 is so rewarding. An extra rival battle and four new Pokémon await those who take their time to explore rather than rushing forward. Come to think of it, I don't think the early Routes have rewarded exploration to that degree ever since. GSC had that southern bit of Route 46, I guess that counts, but still it's strange that the concept was abandoned so early.
Anyway, TL;DR - a greater selection of new Pokémon at the start of the game makes the whole experience better and more varied. Resorting only to the archetypes + old Pokémon makes things feel stale really fast. It says a bit that Cutiefly on Route 2 is the first Gen 7 Pokémon you're likely to meet that isn't a regional starter, bird, bug, or rodent.