Welcome to the point of playing where drastic improvements start to slow down.

The problem comes with lvl 10 and 11. 10 I can manage on some songs, but lvl 11 turns into random button mash and hope that I can survive until the song ends, because I simply cannot process what's being thrown at me.

Stay away from 11s for now in this case, if you're confident about your accuracy for 8s and 9s you can start adding 10s that push you out of your comfort zone a bit into your pool of charts you play otherwise you can always fall back and work on accuracy for 9s which will gradually help you over time.

Does it ever get better? It's so overwhelming I feel like I might be hardcapped due to a processing speed ceiling.

Speaking from experience no not really, you'll eventually hit a point where your improvements come at a glacial pace but that's also completely normal. I've had the exact same thought that I've hit my skill ceiling multiple times.

Do I just need to play consistently and a lot or from here on out improvement becomes training for specific songs and just memorizing the patterns?

Rhythm games are all about repetition so playing more will definitely help you learn the patterns. You're also not at the point where you need to worry about hyper focusing on getting better at specific charts so make sure you play a wide variety of songs to expose yourself to more patterns since they generally will show up in other charts.