I haven't put a scope on all three pins at the same time. Figured it wouldn't make sense for the SPI interface to have more than one ~CS enabled at the same time at any given time.
But I think I found the problem - Noise.
Right now I have hooked the MAX6675 up to three free GPIO pins, separate from the SPI interface, and bitbang to read the temperature. Initially the temperatures sent back were highly erratic (10.5C, 32.75C, 16.25C,...) indicating a noise issue. By coincidence I found out that when I tried measuring what was going on, the temperature readings went stable as soon as I connected the ground lead of my scope to the ground pin, so I clearly had a noise problem. This could also explain the poor behaviour on the SPI interface.
I went through the MAX6675 datasheet once again, which suggested a 0.1uF cap between the thermocouple leads, but this showed no significant improvement.
It also stated that pin T- had to be grounded, so I tried a wire bridge between T- and the ground pin of the tiny thermocouple breakout board (perhaps 2 cm away) and to my surprise the readings stabilised - I'd have expected the board layout itself would already catered for this. In any case I've soldered a short wire in place, which has much improved things. All of a sudden I have usable readings once again, and in addition I've learned how to use the GPIO pins - which is knowledge that'll come in handy in the near future.
Best,
Marc