Can I pull CAT6 cable via old RJ11 panels?

There are two problems you are likely to run into.

1. Pulling new wiring with existing is hit and miss, sometimes you can do it but sometimes wires are clipped/stapled in places you can't access, sometimes the cable routes have too many turns or obstructions to pull and sometimes the wires are just directly buried in plaster.

2. Old phone wiring was often daisy chained whereas Ethernet wiring needs an individual run from each point to the network switch.

The bottom line is you may be able to reduce the damage to decor by pulling wiring in some places using the existing wiring but it is very likely you will end up having to do some damage in the process of getting the new wiring in.

Pulling through the wall between an outlet and a space under the floor or above the ceiling may or may not be possible, pulling through a run between multiple rooms without any intermediate pull points is highly unlikely to be possible.