Why don't aeroplanes use propellers with more blades like the intake on a fanjet?
Every blade will create its own boundary layer and its own vortex sheet. It is more efficient to use fewer blades with deeper chord, because the forward part of a boundary layer contributes most to friction drag.

To keep the lift coefficient on the propeller blade sections in a reasonable range (0.6 to 1.0) for efficiency means that blade chord will be reduced, which will make them less stiff. Again, it will be better to reduce blade count to arrive at a more viable design. All the thrust the propeller creates is pulling on those skinny blades, and they must be strong enough to withstand this force.

Only when the propeller disc loading increases do more blades begin to make sense:
 When engine power increases, the propeller disc area should also grow, but this growth is limited by the resulting speed of the blade tips. Once the flow speed there becomes supersonic, the drag at this section of the blade increases without a corresponding increase in thrust. To avoid that the next best option is to increase the solidity ratio of the propeller, called also the activity ratio.

Make no mistake, this is bad for efficiency. But if there is enough power available, adding more blades is the best way out.

You are right, a lower prop speed allows to increase its diameter, but while tip speed will drop by less than the reduction in prop speed (after all, flight speed should not change), the available thrust from this propeller will drop by the square of the speed reduction, since thrust is proportional to the dynamic pressure on the blades. And thrust you get only from the circumferential fraction of the local speed at the blade; flight speed does not count here and does not help to mitigate the reduction.