How does a connection of a wire to a bus bar work without overheating?
I don't agree that the contact area is all that small.  Copper is fairly maleable, so will deform somewhat when the screw is tightened.  That will make it fit the shape of the hole for a little ways at the bottom, and the bottom of the screw at the top.  Keep in mind the bottom deformation will be a bit longer than the diameter of the screw at least.

If you add up all the contact area, it should exceed the cross section area of the wire.  It's probably at least twice that actually.  If the connection is made properly, then it is gas-tight and should be effectively continuous copper.  As long as it is equivalent to the solid copper of the wire diameter, it will dissipate no more power than the wire itself for the same length.  Even if it dissipates a little more, the bar is capable of dissipating far more heat than the wire with insulation cover.

You could come from it the other way and do the math.  The largest typical house currents are 30 A, which requires #10 cable legally, at least here in the US.  A copper bar like that should be able to get rid of 1 W of heat per junction point without any temperature rise that matters.  The resistance to dissipate 1 W at 30 A is 1.1 m&Omega;.  That would be a surprisingly high resistance for such a joint with a #10 cable.

If you really want to know, test it.  Get a bar, some copper cable, and test the resistance.  It will be so low that you will need to do 4-wire measurement at decent current like a few Amps, but it should be doable.