I've been playing piano for ~25 years. I pretty quickly gravitated away from learning existing pieces towards composition and improv. I'm slow at reading sheet music and can't sight read, but for playing new songs I can generally just look at the chords and get something at least close enough to sing over right away. Just messing around making stuff up is a lot more relaxing than learning by rote I think. Of course I'm never going to play Piano Concerto No. 21 so it depends what you want out of life. :)
The best example of "relaxed piano playing" I can think of is actually the old Build Mode music from The Sims[0]. I'm 90% sure that all those piano tracks are just someone messing around improvising over a few simple chords, because it sounds so much like what comes out when I'm doing the same kind of thing.
Now, it's always a little stressful to record that kind of playing because if you play something bad sounding then you'll have to start over (or at least, splice multiple takes together), but if you're just playing the piano for fun, I think it mostly is relaxing.
Have you ever tried just playing some chords in the left hand and kind of letting the right hand do what it feels like?
[0] Here's Build 3 for example: https://www.youtube.com/watch?v=8gGHQicQUw4
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Edit: Looks like that YouTube video actually links to an article which confirms my old suspicion that it's improv piano:
> "When Russo and Martin couldn't quite capture that wistfulness themselves, they called Burr. A successful jazz pianist in his own right as well as a highly sought-after session musician, he was comfortable working off of Martin's ideas (and, unlike Russo, he was a pianist by trade rather than a saxophonist). Burr says he loved the cut-and-paste style of recording that Martin employed. It was a case, he says, of "just looking at Jerry and asking, 'Does this sound like what you want?'" So, while Russo went away and worked on his pieces only to see them chopped up by Martin, Burr was happy improvising."