It seems that 20 years of not playing a saxophone was a more important factor than the eBay instrument. After three or four practices I can now hit all notes crisply. This may, however, also have something to do with using Rico Royal reeds instead of the reeds that the instrument came with — I have not tested this theory with a back-to-back comparison, and probably won’t.
That does not mean that I can now play the tenor saxophone well — it just means that I can no longer blame the cheap instrument for my lack of musical ability!!