> In a specific cohort lets say age 18-35 - is the only thing that matters for this study "access" to women not how many women a man can married?

No. What seems to matter in the study is whether men have acceptable marriage prospects. (It'd be interesting to see, in that context, whether permitting same-sex marriage reinforces the effect observed and further reduces the same problems. It'd also be interesting to see, as marriage itself becomes less of an expectation, whether the effect is more from men being married -- in which case denormalizing marriage would be expected to reverse the benefits -- or whether the effect is more from men not being alienated by being unable to meet a social expectation of marriage -- in which case denormalizing marriage would be expected to reinforce the benefits.)

By "access" are you just referring to prostitution?