Saying a study is garbage when you disagree with the results is not how evidence based medicine works. Unless there was actual fraud or gross incompetence, every study should provide something of value.
Hypothetically, if vaccination provides enhanced short term protection for people after catching COVID which also provides short term protection then recommending waiting X months after getting COVID to get vaccinated is one option. Alternatively, if post COVID vaccination did absolutely nothing then you could recommend skipping it, ignoring false positives.
1) in retrospect it was obviously incorrect, and
2) even at the time, a reasonable scientist could have concluded that it was flawed
I don't generally call papers "flaming garbage", but some are so egregiously, transparently awful -- at the time of publication -- that saying otherwise is dishonest. That MMWR study was one.
Your only complaint relevant at the time of publication was people that where vaccinated acted differently, that’s also true of the study you linked. It wasn’t double blinded making it “hot garbage” by your logic. Presumably the only reason you linked it was it didn’t contradict your assumptions, even though it failed to address the actual question of how useful post COVID vaccination is.
Except we care about real world outcomes so a double blind study would be less relevant making your original complaint pointless.