Oh, dear, nobody wanted Elizabeth for Barnabas. I've seen two objections to her: 1) consanguinity and 2) resemblance to his mother.
1) Consanguinity: not a problem. There was no objection to a marriage between Barnabas's uncle and Elizabeth's I-don't-know-how-many-times-great-aunt, so Barnabas could marry Elizabeth.
2) Resemblance to his mother: I have three count-'em-3 answers to that.
a) This is something that never occurred to me, because Naomi and Elizabeth are such separate people in my mind.
b) Men quite often marry women who look like their mother.
c) In fact there was only a vague family resemblance between the Naomi and Elizabeth; they didn't look alike. What we see on Dark Shadows is a theatrical representation of the history of the Collins family. Budgetary considerations and an effort to make the story more accessible to viewers compelled Dan Curtis to use the same actress for Naomi and Elizabeth and the same actor for Roger and Joshua, and so on - but it wasn't really like that. Oh, and by the way, the real reason that we never learned the true story of Vicky's parentage was that the surviving members of the Collins family threatened to sued that pants off of Dan Curtis Productions if it was made public.