Is "familial" still utilised in modern English,
e.g. familial problems? Or is it replaced by a noun, e.g. family problems?

See a translation

Familial is used in "The familial medical history indicates a possibility of [ some problem ]" or "Familial issues may be a problem here" (ie. pyschological issues). But you can equally say "Family issues need to be addressed here". I would say "familial" generally means the extended family, eg. are there any examples of this medical problem in the whole extended family, back to grandparents, out to cousins. But if you're only talking about the immediate family, parents and their children, you would say "family issues". Hope that helps.