Tell your husband that we are all pulling for him. Don't be upset at him. Unfortunately, being an engineer who analyzes data routinely, one thing I can legitimately tell you is that you can not prove a negative. We've had the entire world trying to do it for the last year and a half or so, and unfortunately when you take politics and emotion out of it, you still can't.
I am relatively high risk-- like I tick almost all the boxes in age and medical reasons. I took Moderna, had a bad reaction to shot one and none to two, unlike just about everyone else. My wife and daughter did too (daughter with Pfizer). (This likely means that likely we had it before.. Most of my kids and I had a bad flu in 11/2019 with unspecified viral pneumonia at the time, but never had to be hospitalized. Again I can't prove it, and it was just before the official timeline started, so I'd be told I am full of it anyway.) My adult boys refused. Yes the ones mentioned below.
You can't prove the "you will have a milder case" after the vaccine either, as that also is trying to prove a negative. The best you can do is try to do statistical analysis to see if a group is that way (though a control group is needed too), but even then you can't rule out improvements in treatment, whatever herd immunity is actually out there, etc. in better results over time. There are too many dependent events to clearly say this.
Recently, we had a round of what I think was Delta go through our neighborhood and house, and I am pretty sure I got a breakthrough case despite being vaccinated a few weeks ago. It was mild enough that once I tested all four of my kids, two were positive, mine was already negative-- but I had the same symptoms. Maybe a little milder. I thought it was seasonal allergies at the time because it hit at the right time of year for that, but on retrospect it was worse. So, would have it been worse if I hadn't been vaccinated? Maybe? No one can ever know unfortunately. The boys were slightly worse and it lasted a few days longer, but again, nothing to prove here.
We are at the point of being endemic with this virus and actually already at the point of having the spike protein evolve enough to evade the mRNA generated antibodies enough that the current vaccines are legitimately in question at this point. That had never been a point of debate with flu vaccines, so I don't really get the debate with the COVID-19 vaccines. This is predictable.
Don't beat up on him too badly. Respect his decision and hope he comes out of this stronger (which is likely honestly). Nothing is black and white, especially not this. "I told you so" will NOT help your husband.
For the general population Omnicron looks like it might actually be a blessing since it looks like evolution is turning this back into a cold. It might get us to the endemic "herd" immunity we need, if such a thing actually is real.
We will keep him in our thoughts and prayers. Best of luck to you and your husband.