I Think We're Alone Now is a good fit, and also a song I already adore.

Of course, consistent rhythm is very common, but (in the music I listen to at least) it's not often where another instrument takes the role of being the main beat of the song and replacing drums in that job.

I'll listen to the rest of your suggestions later. I'm especially looking forward to Heart and Soul because it's such a great melody but the version in my collection is probably from the late '30s or early '40s so the quality is not great. Thanks!