To put some numbers to that: a 1oz 4-layer 0.06" board will have about 2-4 times more lateral (heat spreading) conductivity when all four layers are mostly filled with copper, versus bare board.
The thru-plane conductivity of bare board is something like 1/5th that of its lateral figure. So, spreading heat out into wide areas first, then through the board, is very effective. Vias are also very effective.
The thermal and electrical (DC) capacity of vias is also about doubled when filled with solder, if you can accommodate that in the design (e.g., large diameter vias in a wave soldering process).
Beware, via-in-pad can be a solderability issue for paste-and-reflow process.
Mind, soldering is basically free-for-all if you're hand soldering and inspecting.
Tim