"Use Apple Watch to unlock your applications and your Mac" setting turns off when Mac restarts

For what it's worth, the following Discussion describes the procedure I used after waiting a few days with no resolution:

Can't unlock Mac with Watch since Ventura… - Apple Community

As a perfunctory prerequisite be sure to follow Apple's steps first, as I recommended in that Discussion, but which didn't work for me: Unlock your Mac with your Apple Watch - Apple Support. "Restart your Mac" is frequently overlooked (but you tried that).

There is no harm in trying it now vs waiting a few days.

You shouldn't have to do that.

What you describe has occurred in one form or another for quite some time. However, it has been my experience that it fixed itself after a couple of days, with no particular action on my part. I recommend waiting at least that long and if it persists, contact Apple and tell them about it. They may have something to suggest, or they may be totally clueless. In any event it strikes me as a bug they ought to know about.

Contact them here: Official Apple Support. Perhaps they will take an interest, but I doubt there is anything we can to about it right now.

The problem used to resolve itself after a day or two of me doing literally nothing. It manifested exactly as you described.

There was one occasion in which it did not fix itself though, and I had to go through some conniptions to make it stick. I'll research my notes and let you know what I come up with, but whatever I did may not be applicable to Sequoia.

I'd give it a day or two before contacting Apple. Nevertheless, it's a problem that we should not have to tolerate, it's been going on a couple years at least, and they ought to know about it.

Interesting.

It's amazing how quickly we got used to that feature, and how annoying it can be when it doesn't work. Thanks for the update!