I had quite the opposite experience ; plenty of people told me they had (mild) flu symptoms - a lot of TMI, to be honest - and assumed they had had COVID already. It later turned out, with no exception, that they absolutely did not.
Scaring those people away would probably hurt a lot, at least in my anecdotal experience.
Do you mean tested for antibodies with finger prick blood test, or PCR random number generator (see elon musk wild PCR results).
I was sick, had test that showed antibodies, and a confirmation letter from my Dr.
Since the EUA to rush out the vaccines does not include those who had it and recovered, it needs more testing and study before it is approved for those who have antibodies. The jab is not risk-free.
I was in a similar position as you: possible early COVID exposure (though no symptoms), several positive antibody tests, but absolutely zero side effects from either Pfizer shot. Annoyed not a few people though by my lack of any reaction to the shots, so that was kinda fun.
And while trying to predict second-order behavior effects is tricky either way it is pretty clear that saying "don't get the vaccine if you already had Covid" would scare some people away.