There are many levels of sha’as had’chak, ranging from slight inconvenience or increased cost, to choleh. I don’t think R’ Moshe only wanted people to rely in the heter for a sha’as had’chak on the level of choleh where there’s no alternative.
However, the fact that R’ Moshe intentionally (according to R’ Belsky) made sure to add “baal nefesh yachmir” to every teshuva, plus feels that mosdod hachinuch should be makpid even at great cost, plus, despite his great anavah, writes that he personally was makpid, is a strong indication that where there’s no or little extra cost or inconvenience, he feels one should not rely on the heter.