Are there additional benefits to fielding max-rank soldiers?

Solution 1:

As far as I know, once your soldiers reach Colonel rank, they no longer advance (unless you count Psi training as advancement, but as far as I know, that has nothing to do with killing aliens in the field). The only advantage of having elite soldiers is the fact that you have elite soldiers on the field - more powerful, higher HP and more perks.

Solution 2:

They will not improve in any way unless they are Psi-gifted and haven't yet advanced down that path (then you will have to field them, to improve them - not via kills, but by using their abilities).

Still, if you play on Classic difficulty, you will rarely have the luxury of leaving your best soldiers at home. You'll need them. Desperatly. Missions keep surprising you with tougher enemies (or even just unexpectly bad RNG-numbers). Good squad members can be the difference between one or two deaths and a wipe.

Usually the downtime from injuries will be (more than) enough to train other rookies. The game sort of forces a rotation on you anyway.