I thought that serious Jews didn’t accept converts anyway (because their relationship with God was specific to certain bloodlines).
RE: because their relationship with God was specific to certain bloodlines
Would Moabites and Jericho-ites be considered acceptable blood lines?
Serious Jews DO accept sincere converts, as a matter of religious obligation. Jewishness is a (religious) legal status and not related to DNA at all.
By the way, something stated in the Talmud is no way determines Jewish law. The Talmud is a record of discussions and opinions which may or may not be a matter of Jewish law.
“I thought that serious Jews didn’t accept converts anyway (because their relationship with God was specific to certain bloodlines).”
The opposite is closer to the truth. The Torah repeatedly exhorts us to love converts and to take great care not to make them embarrassed for having been born into other faiths. A famous example, of course is Ruth, for whom an entire book of scripture is named, telling the story of how she converted and ultimately merited to be the great grandmother of King David.