It's meant to be a call back to the days of "Lincoln Republicans" - when President Lincoln built much of his reputation on a true "rags to power" story after being born in a literal log cabin in Kentucky. Lincoln Republicans coalesced around Lincoln's belief in true equality and freedom for all regardless of race. The LCR is an extension of that and arose during a fight against anti-LGBT laws in California in the 1970s.
However I'd argue that the rest of the GOP has fallen so far down the rabbit hole these days that the LCR's are "in name only" supporters of gay rights and have no reason to exist as an organization.
That's an interesting article I hadn't seen, thanks. And while I doubt anyone would be able to say definitively one way or the other at this point, from that article it sounds like the evidence that he was homosexual or bi is...pretty thin. At best.
It's all fluid anyway... He could have been pansexual, or demisexual (or maybe even something else really cool that I don't even know about yet). As far as I know Speed was his only possible known male suitor.
Please note: I'm not even close to an expert on Lincoln, and this comment was based almost entirely on memory of when the History Channel used to actually show historical documentaries (plus maybe a little Drunk History mixed in).
They refer to the idea of closeted right wing gay men who have marriages in public, but do retreats in secluded log cabins together to do gay sex and stuff. It's a log cabin because they are traditionally seen as secluded weekend houses.