Which is why I like my Yamaha Clavinova - not only does it sound great (and never has to be tuned), can play with headphones.
PS I do the same with electric guitar
If you're seeking something that feels like a piano, the Clavinova is probably the closest electric piano...but it's still not really close to the real thing. There is no substitute for the key throwing a hammer that actually hits something.
If you can deal with the sound/not having anything but an (electric, not acoustic) damper, the Fender/Rhodes 88 is the closest I've played on, and I've been on everything from the cheesiest Casio all the way up to a 13 foot concert grand.
I'm legitimately surprised that they weren't/aren't more popular than they are.