“converts were just taught about a few mitzvos before completing their gerus.”
Right. But they fully accepted and committed to honor and carry out all mitzvos as they became aware of them.
“Instead, what probably matters is the ger’s intention at the moment of intention — did they intend to accept the yoke of all the mitzvos?”
Exactly. So if they’re joining a community that doesn’t adhere to certain mitzvos, with the expectation that she too needn’t adhere to those mitzvos, i.e. eating non-kosher, immodestly, only “half Shabbos”, etc., then her intention at the moment of immersion is to ignore and not keep the mitzvos that her Jewish fiance and the Jewish community she’s intending to join do not keep either. Thus her intentions cause the conversion to not able to be a valid one.