I'm of the belief that, when a puro company doesn't have a gaijin that is a legit star, they can live with having their top foreigner not being the main champion or being in contention all the time. So to me, top gaijin doesn't = AJ Styles/Kenny Omega spot. (other fans don't see it that way though, that why I wrote that was part of the backlash against White).
When Karl Anderson was the main foreigner, dude had some huge matches - were he did great btw, he had a low key very good run fo a couple of years - but he wasn't the 1B star of the company. He wasn't forced into that position. Jay White, nor any gaijing on the roster right now, HAVE to fill that spot right now. That's my main issue with it to be honest.
It feels like New Japan is forcing White to a spot that neither he nor NJ need at the moment. I think dude could be a legit player/star/draw for them in the near future, just fukking wait till he gets there. When Omega left (and he'll be back eventually), he didn't left a void that had to be desperately filled by JW, ZSJ, Ospray, or whoever.
So to the question "if not Jay White, who?". My answer is: right now, nobody. And that's fine. Wait till someone gets enough of a reaction that can justify that level of push. At this moment, it feels like Moxley could be a legit player for New Japan if the AEW relationship gets sorted out soon. But if that doesn't work out, then just keep waiting. Give Jay White other stuff to do instead of constantly being in contention, there are a ton of stuff dude could do to build up to a run on top that actually works. There's no need to rush anything.
BTW, I understand why Gedo did this with White. He basically tried to apply the same formula he did with Omega, which was a huge gamble Gedo made and that paid off better than anyone could've foreseen. I can see the reasoning behind doing it with a "home grown" dude like White.