ujm: The issue of safek goy, safek mamzer arose recently with the Ethiopian Jewish community, and there seems to have been a consensus reached that a nominal conversion would work to remove both safekos.
In America, where someone with non-frum ancestors applies to go to a frum school (not to mention, if they get married), the standard practice has been to chat about their ancestors enough to know that they can trace their ancestry back to someone frum, and to be able to determine if there are any issues. At this point the question is only about a few generations (excluding the early 19th century Germans, and the colonial era Sefardim). In the foreseeable future that will be very hard to do, especially with families that had been non-frum for four of five generations, in which “kiruv” for all purposes evolves into a “conversion” movement. It also should be noted that already the percentage of Americans with Jewish ancestors is rising steadily so that increasing we are in a world where all goyim are safek Jews (perhaps an argument against use of a Shabbos Goy).