I think that first off, you need to decide if you want a keyboardist or a pianist. My experience has been to watch out for pianists. They are often used to playing by themselves and will tend to try to cover everyone's parts... melody, rhythm and bass. The joke of tying the pianist's left hand behind him is actually not a joke. If the band has a piano oriented, that may work OK.
I would much rather work with a keyboardist who is willing, when needed, to strip his parts down to just what is needed to fill in the holes, often using a patch other than "piano", i.e. horn parts, organ/synth pads and that kind of stuff. Make sure you get a team player.
I would agree with part of this. The decision for me is "pianist, organist, keyboardist or combination player (who probably doesn't solo well but can play pads and other soundscape-type stuff.)"
There's a wide variety of uses for keyboarding instruments and what each band needs is different.
There's no reason a piano player can't play correctly with an ensemble. If you hire someone who is used to playing only SOLO piano pieces, you might have some problems around their left hand. (And, I hate to break it to you, but if they're good readers they're probably playing the bass lines correctly as written and if you aren't doubling that well, you probably aren't. But I digress.)
But any good/experienced ensemble player knows what to play with their left hand to stay out of everyone else's way. The "truth" in the old saying is for inexperienced players.
That said, I'm in one band that has a great piano/organ player and we call her our secret sauce because she's excellent at comping through chord changes and playing solos in R&B type, improvisational situations.
I fired her from another band because she really didn't have the time or the will to learn real parts. You can't play serious covers of more contemporary music with someone who just comps through changes. Parts have to be learned. So out she went.
Another band I'm in has two keyboard players. One is a great soloist, the other is a great solo PLAYER in the sense that he's used to doing those one-man gigs where he covers everything. He's more of a pianist and not so good at improvising.
We mostly have him playing pads and filler stuff so that they stay out of each other's way. None of them ever get in my way.