Getting a letter of recommendation from a socially controversial PhD advisor
While there is always risk in such things, if his professional credentials are good and the other comments don't reflect badly on that, I'd suggest asking for the recommendation and not otherwise mentioning it in cover letters and such. Treat it as a purely professional relationship, which it should be.
As you note it is odd to not have a recommendation from one's advisor. And I'm assuming that your judgement is that his letter would be professional. If you have a way to find out anything about what he has written for others, it would be a benefit. A department head might, possibly, give you some guidance. But that is another judgement call - whether to ask.
Of course, you can't control the reactions of others, but you can, if asked, disassociate yourself from his views. Just say that you don't hold those views in any interview or such.
I'll note that in math and cs there have been a number of similar situations where a person is widely respected professionally, but whose views on some issues have been shunned and the people condemned. It doesn't reflect on their technical work, not should it reflect badly on you or your work.

See the case of William Shockley, and that of Robert Lee Moore.
And, of course, many people hold appalling views of things, that may be known to others, but not to their students.