This app was mentioned in 8 comments, with an average of 1.38 upvotes
I'm in the same situation as you (first tablet), and I agree with all your points. I'm still running stock AOSP.
I was not happy with the memory leak in 5.0, the constant redraws, etc. but with 5.1.1 it seems to be mostly gone.
I also replaced Chrome with Mercury Browser as it seems there's no way for Chrome to keep tabs in memory when you switch between them... it's just ridiculous. Mercury works just fine.
And in fact I'm so happy with it, I purchased this keyboard... I'm so satisfied with it. It integrates perfectly well with the Nexus 9, all shortcuts work seamlessly (CTRL+C - CTRL+V, ALT-TAB to switch between apps, shift + arrows to select text...). I'm surprised at how much of Android can be navigated just using the arrow keys! Miles better than the iPad Air I tested it on this weekend.
I use it a lot for work email, book reading (play books is AWESOME) and watching TV series, as well as web browsing, chatting in Telegram, etc.
I'm really happy with it, and indeed the UI is much, much better than other tablets I've tried (Samsung Galaxy Tabs, iPads, etc.). I keep trying any tablet that falls on my hands (especially at stores, etc.) but I certainly wouldn't swap it for any other right now.
No votes for Mercury Browser? For me, it's far and away the best. The ad blocker doesn't really work, but everything else is just brilliant. It's a very fast browser, and has lots of configurable gestures for just about anything you can think of. You need to dive into the settings and personalise it when you first run it, but after that it's the fastest and most enjoyable I've used.
This browser allows user agent switching which might help: https://play.google.com/store/apps/details?id=com.ilegendsoft.mercury