It does seem the 13 amp plug has been down graded, the necking of the live pins to put plastic sleeves on them has reduced the ability to transfer heat away from the fuse, black plugs are better than white, and clearly need to be in free air, the whole design of the ring final and 13 amp plug during the war was with the idea of powering heaters, so they should be OK, and with a thermostatic controlled oil filled radiator I have not had a problem, but 3 kW does seem today to be two much as a continuous rating. So as to if 20A switch or 13A fused spur depends what feeds it, from a ring final would need to be fused spur and with the down grading of cable from 7/0.029 (2.9 mm²) to 2.5 mm² one needs to be careful drawing over 2 kW from a ring final, OK near the centre but near the consumer unit can cause one leg to be over loaded.
The IET BS 7671 now warns of the problem with over 2 kW fixed appliances and suggests over 2 kW should have a dedicated supply, which we have done for years with the immersion heater, but do not tend to have done with tumble drier or washer drier, there has been a move to 2 kW for cloths drying, but it means the answer to your question in not cut and dried due to the down grading of the ring final.
Least we forget the ring final was designed during the war to assist with the rebuilding they realised would be required post war, and the shortage of copper. It does seem a FCU can dissipate the heat from the fuse better than a plug.