Icelanders have taken to using a phone app to check their date isn't too closely related (if any country needs immigrants urgently, it's them!!) and ancestral research is getting popular, so the technology to ward off most of the relatedness difficulties already exists.
It is analagous to that thing where if you count back far enough the number of theoretical nth-great-grandparents eventually exceeds medieval Europe's population so, in theory, Charlemagne's genes will have mingled with just about everybody's family line by now. (Okay, it's a thought experiment only but you het my drift).
Of course if one donor's attributes are so popular that hundreds of women in a small geographical area select him, then there will be a cohort of children of a similar age who are at serious risk of pairing off with one another.
In this sense, parentage involving only having handfuls of progeny, who know they are related by means of shared upbringing, is a strategy which avoids inbreeding. An actual harem system (eg deer herds) or something which approaches it (sperm bank) contains the inherent risk of inbreeding over a very short timescale. Making donors traceable empowers the child to build
a proper family tree and thus avoid marrying siblings, cousins etc.