It would dilute the significance of that Red fight. The concept of the Red fight isn't just a good concept, it's practically a necessity for me to consider HGSS good games personally.
No, what dilutes the Red fight is them using him over and over because he's the ONLY former protagonist you can battle as an NPC. Now, this probably wouldn't stop the reuses of Red, the only instance it possibly would may be for Sun & Moon (and even then possibly wouldn't prevent it cause 20th anniversary).
To be fair, context is important, you can't just toss in a previous game protag without there being a reason. To be fair the only immediate "logical" other protag fight would be the BW protag in B2W2 that was planned & scrapped. All the others they would have to come up with a workaround reason (if anything at all). Also, the timeline is important here as we can't have Ethan/Kris & Lucas/Dawn appearing in the Kanto or Hoenn games, and from then on in remakes we can't have protags from future games show up (the only exception being B2W2 as they supposedly took place at the same time as XY, but then you'd have to come up with a reason why Calem/Serena took a trip to Unova. Of course, I say all this while Sun & Moon has Cynthia and Wally for no particular reason show up in the Battle Tree. Say, how about instead of those two, we included the previous protagonists. Heck, considering in Sun & Moon the player becomes Alola's first champion, Red (& Blue) run the Battle Tree, and its an anniversary game why not have had at least one protag from every region appear in the Battle Tree. Wouldn't even need to explain who they are aside from a Champion from another region. They wouldn't even need to make official teams for them, just give them a batch of good mons from their generation to pick randomly (maybe making sure they pick at least one of the Starters) and there ya go!
Also, if having other protags being battle-able in other games somehow takes away your enjoyment of Red being battle-able in HGSS, I gotta ask: Are you SURE you like the Red battle in HGSS (or Johto games in general)? Because it sounds like you actually don't, you only accept it because its the only former protag that can be battled, which isn't a good way to look at it.
Johto and Kanto have a unique dynamic amongst regions whereby Johto is almost like a little brother to Kanto. (...) Kanto is a fair amount larger than Johto in both size and population based on the Japanese regions they're inspired by. Figuratively, Kanto is much more modern and cutting edge while Johto appears behind the times in comparison.
Pokémon wise, Johto is pretty clearly framed as inferior. Many Johto Pokémon have Kanto ties in the form of their evolutions, and the number of Johto Pokémon count 33% less than Kanto. Battling wise, Johto Pokémon also fall short. Half the Johto gym leaders using Kanto aces speaks to even the Johto natives believing their native Pokémon aren't up to the task of living up to Kanto.
Additionally, Johto is the only region which does not have its own Elite Four. Some may claim Johto and Kanto share an Elite Four but I think it really belongs to Kanto. You're literally in Kanto from a map perspective once you're at Indigo Plateau. Yet another example of Kanto overshadowing Johto.
I think you're seeing things a bit too literally. The problem with the way they have presented Johto is that it didn't exist when they were making the first games. In-lore, the Gen I Pokemon in Johto are just as native as the new Gen II Pokemon (heck, they also have Gen II Pokemon appearing in Kanto including some exclusives (though that was probably a bad idea in the long run)). They're on the same landmass, they share the same shoreline, Pokemon from one are going to cross over to the other. It's not strange for half the Gym Leaders to be using Gen I mons because to them they are using native Pokemon, it's just out of meta the player saw that Pokemon a few years ago in the original Gen I games.
Johto isn't behind the times, its meant to be a counterpart to the more technologcial Kanto by focusing on traditions and history. Now, it could be because we're not native Japanese we miss a lot of these cultural reference, but to say they're "behind the times" is shortsighted to what they're trying to do with Johto. If Johto was just as technological as Kanto it would feel even more like just an expansion pack.
But yeah, a lot of Johto mons fall short. I mean, who uses Azumarill, Wobbuffet, Heracross, Sneasel, Skarmory, Donphan, Smeargle, Miltank, and Tyranitar? LOSERS, that's who. These Pokemon can't stack up to the likes of Butterfree, Beedrill, Raticate, Arbok, Wigglytuff, Vileplume, Venomoth, Dugtrio, Persian, Golduck, Victreebel, Golem, Rapidash, Farfetch'd, Dewgong, Hypno, Exeggutor, Lickitung, Tangela, & Seaking; all works of art and masters of battle. And how dare Johto defile Gen I Pokemon by giving them inferior evolutions like Crobat, Steelix, Scizor, Kingdra, & Blissey; without Johto they wouldn't exist and all their superior prevos would be better off for it!... This is sarcasm BTW. Sure, all Gen II Pokemon aren't winners, but neither are Gen I. Also should the number of Pokemon really be held against Johto? Okay, it has less Pokemon... so what? You wanted 150 new Pokemon and only them in the Johto? While I give crap to GF for sometimes not thinking ahead, I'll give early GF credit for thinking ahead about not only restraining the amount of new Pokemon but also using older Pokemon to fill out regional dexes cause there's nothing wrong with the older Pokemon so are vital tools.
But you're right about the Elite Four belonging to Kanto which is why it's located deep within Kanto, they don't accept the Johto Badges and they kept the Elite Four exactly the same as the Gen I games. Oh, wait, it's inbetween both Regions, they do accept the Johto Badges, and the only returning Elite Four members are Bruno and sorta Lance though he's now Champion; Lance who btw is from Blackthorn City which is located in Johto so the current Champion of the Indigo League is a Johtian (and the new "top" of the Elite Four uses the new Dark-type that was introduced in Gen II).
Also, could it also be a reason you feel Kanto is superior is because it was made the post game area thus every trainer there was given a higher level in order to keep up with the player? It's personally why I would like to see a game which starts out in Kanto but then lets the player go into Johto, to see how the Johto region adjusts to being the post game area that has to have the higher level trainers.
Which is why I think the underlying theme of GSC and HGSS is that it's up to YOU as the protagonist to carry the torch for Johto and prove that it can compete with the vaunted Kanto region. It's like an underdog story in a way. Which is why I try to use only Johto Pokémon on my GSC and HGSS playthroughs, despite how difficult that is to do in these games, since it's thematically significant to me.
And it's also why, again, I believe the Red fight at the end of these games is a necessity to close the narrative and prove that Johto has finally arrived, by beating the greatest Kanto trainer. No other pair of regions has this kind of dynamic which is why I don't see the point of implementing such fights with other protagonists in other games. It would only serve to dilute the significance of the one time where it makes sense.
I just want to note that you have an interesting perspective of this and I can see where you're coming from. It is certainly Johto showing they're as good as a region Kanto is... but I also feel that's done more through exploring the region and seeing what it has to offer than a hero of Johto coming to kick the butt of the previous protag. Because, he's a problem with that: we're no longer playing Red. Now he's just another NPC, controlled by the computer and, while made tough, was also made to lose just from the fact his Pokemon are locked. While if you lose to him you can go back and train, Red will never change his team, raise their levels, or change their Move. It's only a matter of time before you defeat Red. And, as I noted, the Champion is already a Johtian, if anything Johto has proven itself capable and its more Kanto is sort of stagnating. Though, once again, is it actually stagnating OR is it just that way to the player because we played the Kanto games so we know all about Kanto thus only the new stuff and what's going on with the characters has any novelty.
You're right, no other regions shares the dichotomy of Kanto & Johto, but that shouldn't mean characters from different regions can't interact with one another. Also when one of the core concepts of Pokemon is "being THE master" which seems to indicate being the strongest trainer, well, why wouldn't the other player characters go out looking for the other/future player characters?