I'm not close enough to this situation, but the evidence I have is:
(1) He said so.
(2) There weren't any allegations of improprieties in the past decade or two in the articles I've read.
(3) 30 years ago, likely a majority (and certainly a near-majority) of people had at least some level of discomfort with women serving equally in the military. Today, it's an extremist view held by a tiny minority. Most people who held those views 30 years ago did, in fact, change views, not closet them up.
What you're describing is a symptom of exactly this cancel culture. People DO learn to hold their more objectionable views close to the vest, and that's a problem. People can't actually change views without open and critical conversations.
I live in an extremely progressive city with a lot of racism and bigotry simmering just below the surface. It's bad -- many people in positions of power hold extremely bigoted views. That never plays out in the open, but those people act on those views, either without articulating them, or articulating them as abstractions.
I see no way to address any of that without honest and open conversations, which we can't have. Saying the wrong thing leads to career death, so everyone says the PC thing.
And when racism happens, in most cases, people depart quietly, and move into a similar position at another company / school / police department / etc. It's pretty rare that anything goes public. But if it did -- someone was closed out of the economy because of a perception that they were racist -- what do you think the result of that on racial tensions would be?
It's important to have systems to address and resolve problems. If you have a racist, the desired outcome should be that in a few years, they're not racist anymore. Start there, and work backwards to how to build out systems to do that.
There's also a longer post about the value of due process, and of innocent-until-proven-guilty (which is not the same as NO process, which is what we often have right now).
As a footnote, your discomfort is not the paramount issue here. We have laws to protect former felons, people with bad credit records, etc. from employment discrimination. Even if I might rather not hire a former felon, or someone who can't manage their finances well, I'm not allowed to ask those questions in a job interview. That allows people who've made mistakes -- often much bigger than this one -- to return as contributing members of society. That's a good thing. Otherwise, we end up with revolving doors to jails. Indeed, I'd argue those are the laws which ought to be expanded -- they're not nearly strong enough, and that disproportionately impacts vulnerable populations. We want paths to remediation for everyone, rather than for no one.