The simple answer to this question as to whether you need a good sense of rhythm in order to be able to paly the piano is a yes. To get started though, you do not need to have a good sense of rhythm at all, because a good sense of rhythm is something that you can develop over time.
Trust me when I say you don’t need to have any sort of sense of rhythm because it can be taught. I have taught so many piano students that started out with a less than adequate sense of rhythm. In fact, I would have to say that a natural sense of rhythm is not really common, as the majority of my students over the years have struggled with a poor sense of rhythm, but has always ended mastering what they needed to master. Notice that I have said that it is the majority of my students that can struggle with the rhythmic components of a piece of music.
I myself, was not born with a natural sense of rhythm at all and this was something I wrestled and struggled with for about 5 years, before it became easy for me and the only reason why it became easy for me is that my piano teachers all gave me different tools to help me figure out any rhythm and then be able to practice it until it became so-called ‘natural’ to me.
What you do need is the sense of a beat. A beat is not as complex or specific as a rhythm, but a beat is nevertheless important. If you can clap along to a piece of music – any sort of musci, then you can certainly keep a beat.
I’ve heard that some people can’t keep a beat at all, and while I believe that this is true, I think it is the exception rather than the rule. I can honestly say that I have never had a student who has not been able to feel the beat of a piece of music and if I haven’t taught over a thousand students, over the decades, then the number would be pretty close to it.
So if you are thinking of taking up the piano, then don’t worry if your sense of rhythm doesn’t really exist, because you can learn this. What you do need to have is the sense of a beat, which you would almost certainly have.