At 1oz thickness, 100% coverage of copper foil about doubles the lateral heat spreading capability of the PCB.
At 2oz and 4 layers, that's a factor of 8. Big win.
A dense grid of vias (typically plated to about 1oz) about doubles the through-plane conductivity of the PCB (which, for plain, unadorned FR-4, is about a tenth the lateral conductivity!). Filled with solder, tripled. Big win.
If that's still not enough, a thermal pad between PCB and mounting plate is an excellent strategy. Choose a material with reasonable conductivity (>1W/m/K) and soft durometer, so it conforms around small; chip components. (Avoid using tall components on the PCB's heatsink side.) Thermal pad tech is awesome these days -- at least as good as PCB material, but flexible and insulating (do watch out for partially conductive varieties, though).
Pro tip: cut the thermal pad into strips, leaving gaps inbetween to give it somewhere to flow. Design it to compress slightly under the PCB (by a few thousandths), taking into account the volume of any components placed on that side.
3oz boards are okay for 0.65mm pitch leaded parts, and may be questionable for 0.5mm pitch and leadless parts. I don't think I would go over 4oz for anything that needs TSSOP/SOIC and power devices on it.
Board fabs cannot make vertical sidewalls. The sides of copper traces are always tapered, or lumpy in some way (depending on the exact sequence of etch and plate used). Thus, the minimum trace width/space goes up proportionally with foil thickness.
There are some "super heavy copper" processes out there, which may involve milled copper plate, by the looks of it (milled, then stuffed into a laminate, and more glued on top). The vertical edges in such a process would be good (as long as you don't expect the gaps or webs to be too thin, which would be impossible to machine). But I've never quoted, or seen quoted, such a board...
Tim