The NHSSB has its flaws; one being that, given it was sampled from the US population at large, when you try to look at small minorities (and minorities within minorities) the data starts to dry up. It does have the breakdown by sex data you mention, though.
There are other signs that point the same way. In 2019 a Canadian survey on open relationships (which did not look directly at sexual orientation) they found a remarkable gap between the % of men and women who report such a relationship would be ideal: 18% of men and 6% of women. -- https://pubmed.ncbi.nlm.nih.gov/30932711/
The data on number of sex partners is telling too.
From a recent US population survey, the percent who report 3 or more sex partners in the last 12 months: 35% of gay/bisexual men, 15% of straight men, 6% of straight women, 17% of lesbians. -- https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7293001/
A slightly older Australian study from the early 2000s is a bit more dramatic. Asking the number of partners in the previous five years, it found that for men who identified as homosexual, the mean was 32 and the median 7 partners, with the max capped at >2000 to stop skewing the numbers. For women identified as homosexual, it found a mean of 3 and a median of 1 partner with a maximum of 54. -- https://pubmed.ncbi.nlm.nih.gov/14696706/
Unfortunately all such studies tend to suffer from at least some of the same flaws you mention, with either sample size, reliability of sampling, or issues of identity confusing the situation. (In the Australian study, it seems women who reported multiple same-sex partners identified as heterosexual relatively more often, for example, which follows the pattern of what you noted.)
The particular numbers vary from study to study, but this general pattern is almost always there. While the median number of partners among gay men tends to only be a bit higher than among straight men, the mean is always outright hilarious. A large minority of gay men are quite promiscuous, and there just is not an equivalently-sized phenomenon with women who have same-sex relationships.