Probably not. Its the community, and the amount of information that's here on the forum that is the main draw. You generally get used to any software you use after a certain amount of time, and groundbreaking features or even a forum redesign isn't likely to change much in the statistics.
If there was more competition to Bitcointalk, then it might have a little more effect, but currently I don't think that is the case.
When I first came here, I didn't see the appeal of the staying for long because of how bland it looks. I recalled it was due to locating some recovery wallet phrase for a Bitcoin wallet and setting up some miner's config file and it showed up on first on the SERP.
But competition? The OGs started their journey here, so Bitcointalk but nowadays people are just looking for contents. Legacy, maybe a second or two of marveling but that's pretty much that. Not like we'd see Satoshi and Laszlo returning anytime soon, if ever.
In other words, Bitcointalk was the launchpad of Bitcoin but nowadays people don't really care much about this and would rather focus on making money/ talking about latest developments.
I doubt that new forum software will attract many new users. Design of forum and software isn't main reason why people signup on forum. Main thing is always content. Yes, some people will start using forum because it will be mobile friendly, will look better and will have more features. But number of such users won't be significant.
It's no longer the early 2000s though, when UI was still in its infancy stage.
People want accessibility and now website design has come a long way. Gone were the days when redesigning a website took at least half a year, because now there are tools catered for that. The development process has been simplified over years via trial and error. So budget is definitely not an issue here because there are developers willing to take on such project, it's just if the admin(s) want it or not.
Should theymos really desire for one, pretty sure he'd get it done dinosaur ages ago. But I guess he doesn't want to prioritize if because he's pretty much content with the "as long as it's broken, why fix it?" mantra. After all, money's still pouring in from advertisers and the funds held could feed the entire Africa for at least 10 generations (okay, was exaggerating but you get the gist).
For content, diversity within the community is important to maintain a longer attention span on the website. That's why community discussions like Reddit or Quora thrive because I could be talking about movies for a moment, followed by crypto the next. And most of the replies there are more genuine (not forced), and I'm pretty sure most of us can't deny that.