I think where I sit right now is to spend the rest of the season and winter working on my flexibility, ride for the season, set up the trainer for the winter and then look into a bike for the Spring. Question becomes do I start out small with a $1000 bike (I'd likely want a minimum of a 2x9) or do I invest a little more ($2000 for a 2x11) with the plan that it'll last me longer. Issue I see with the cheaper bike is less "room to grow," but obviously if I don't stick with it the $2000 is a larger drop in the bucket. But that's part of why I won't buy until Spring. If I can't keep on the gym stationary/my trainer at least 3ish times a week, then it likely doesn't make sense to drop the bigger dollars.
If you're thinking about a new bike in spring, you're putting the cart before the horse a bit, but it's always fun to think about getting new toys - sometimes more fun than getting them. Regarding entry level vs midrange: go on youtube and search "gcn midrange vs superbike," and watch that first video. My takeaway was that while the more expensive bike is faster, the differences are marginal for someone who isn't truly racing against the clock. GCN even has an episode where they take a 20yr old Trek and upgrade it to the level of a modern, $1500-ish bike - and again, while the difference with the newer, more expensive bike is there, you're already at the point of severely diminishing returns once you have a bike that fits well, has proper gearing, and is already relatively lightweight.
That said, you are unlikely to find any material outside of aluminium in the $1k price range, but you'll start seeing some entry level carbon bikes around $2k. While the difference isn't as dramatic as it was in prior decades, carbon is known to to have a more 'muted' ride quality that dampens road vibration (think: chipseal), whereas aluminium has more of a reputation for beating up riders a bit more. Of course, that is by reputation only - people who have ridden Cannondale's CAAD12 alu frame swear that it's incredibly compliant, and I can see something like a carbon Colnago CR-S feeling quite rough after a while (granted, my experience on that bike was limited to a 10 minute test ride). When you're ready, go out and start test riding/renting different bikes - it'll tell you a lot.
TL;DR version: You're not likely to be much faster on a $2k bike vs a $1k bike, but spending more opens you up to more options, some of which may be carbon, which often rides more comfortably than aluminium.
Lastly, given that you're 6'3" and thinking about getting out there in spring, you may be able to find last year's leftovers for a song - probably harder to do if you were looking for common sizes in the 52-56cm range, but in the ~60cm range you're looking at, someone might have a 2018 or 2019 beauty going for a song.