The 120 V puck lights tend ot have a very short bulb life.

The day to day vibration of the shelf they are mounted on kills the weak filaments very quickly.

There are a number of vendors that sell 12 V puck lights with small power supplies that also can fit under a shelf and are 30 W (60 W for a dual unit).

I'm trying to envision how best to run the wire. I can easily extend the NM cable up the inside of the wall, so I'm thinking of wiring up the dimmer switch at the backsplash and then bringing the feed up and out to the top of the wall cabinets (they don't extend all the way to the ceiling) where I can place a box for the direct wire conections to the lights below (chasing the wires back down to the pucks inside the wall cabinets). Any issues there?

"I can easily extend the NM cable up the inside of the wall"

All splices for ermananet wiring must be in junction boxes that remain acceaable.

Lighting id also not aloweedf on the 20 amp small appliance branch circuits for the kitchen counters.

All I can say is those crap units are almost impossible for anyone to install to code, or IMO safely.
At least the low voltage ones could be made legal, those junk 120v things are useless.

OH, and this area you describe IS in the kitchen.