Acceptance is support. [and it's exactly what they 'claim' to want] But you're right, most are too emotionally involved with the subject to ever change their minds.
OK, but you accept and therefore "support" sinners of all stripes. We all do, at least in how we interact with the world. One easy example are single parents with children born out of wedlock. Not only does society accept them, we provide them tax benefits. There is no law preventing a cheating husband from dumping his wife and marrying his mistress. Newt did it twice. You may not approve of his choices, but there is no push to make what he did ILLEGAL. We all accept divorce, despite the vows many of us make. Our society
celebrates raw greed and selfishness, so long as it's tied to business. You accept atheists or agnostics in that I doubt if you even bother to inquire about the spiritual health of them when you're on the operating room table. And even if you grill your retailers and service providers and only do business with the righteous, society demands nothing in that regard. And certainly the non-believers or fake believers all get the same access to the rights and obligations of marriage as the devout and "true" (versus the "in name only") Christians. No one inquires about that when the marriage license is signed.
Point is I don't think gays are asking for your spiritual blessing on their lifestyle any more than the agnostics or atheists are, nor am I. My beliefs on homosexuality are the last thing on the list of things to worry about come judgment day. If that's at the top of yours, or in the top tier, great. Your life, your beliefs, your choice. I can respect those beliefs and reject them same way I respect but reject Islam or Buddhism.
But if you start tying government benefits to how well a person lives his or her life in accordance with
your interpretation of the Bible, you'll find that those efforts will be overwhelmingly rejected. But when we do the same to gay couples, they're supposed to accept that without a peep of protest. So when you say that acceptance is all they "claim" (whatever that means) to want, it's not your spiritual "acceptance" just that they be treated like adulterers, the greedy, the divorced, the liars, the non-believers, etc. who (e.g.) can all line up and get a marriage license.