There are multiple versions of the 1962 missal, so it depends which one you're talking about. If you're talking about the original edition, this one did NOT add St Joseph to the canon. I'm not sure if it deleted the 2nd confiteor prayer but most Trad priests add that back in anyways.
The Holy Week changes are also lumped into this discussion but those were in use before 62, and only affect 3 days, but not the actual mass for the rest of the year. The big change of the 62 missal is to the calendar. Many vigils were deleted but also the calendar feast days were simplified. The changes to the calendar are not uber-liberal, even if they aren't Traditional.
Fr Wathen said that he saw no major issues with the 62 missal (assuming you don't add St Joseph to the canon and you add back the 2nd confiteor). He said it wasn't the best but there was no danger to one's Faith. In other words, there are bigger fish to fry.