I was mostly talking about the SMB overworld theme. There were still issues on other ones, but they were faster songs so it was harder to notice. It's always easier to keep a rhythm if it's faster, the SMB theme is not only slower but doesn't really have a constant "backbeat" part, so that tune is where his true skill (and any lack thereof) in keeping time was shown. On other ones like the underwater theme he does much much better, reason most-likely being because his left hand has something constant going that keeps him on track.
What I said is very subjectively interpreted, I'm not saying he's incompetent or anything. And saying he doesn't have a good sense of rhythm is not saying that he has a bad sense of rhythm, just that he has some ways to go. I'm a musician (guitar) and I've had a good amount of experience playing with pianists. I can tell by the way this guy plays that if we were to play together, he would need to listen to me in order to keep his time. He would be able to do so, so he doesn't have bad rhythm, but he probably doesn't have good rhythm if he can't keep it by himself. Very much like my earlier days where I could only play songs on guitar that had my arm in constant motion.
Don't overstate my critiques and understate my praises just because you are very impressed by his performence. I've been doing music for a long time and stuff REALLY stands out to me (lets just say I almost always agreed with Simon 100%). In music generally you don't just want to do it right, but you want it to be so natural to you that you can even subtly add in your own personality and feeling to it. Music without feeling is like food without seasoning, it may get the job done but I'm not going to praise it a whole lot. Technical skill shows your hard work, but it doesn't show your gift.