Under-reporting of Lyme actually shouldn't impact a finding of no link as as far as I can tell. Despite the under-reporting, we still know where Lyme is more or less prevalent in the US, and I believe there will be very big differences due to climate and where ticks who carry Lyme tend to live. If prevalence of Alzheimers doesn't correlate with those big differences, that seems pretty clear.

(edit) Thinking about it more, I think the anecdote you mentioned is a bit different from the parent posts. The anecdote is about a misdiagnosis (he only had Lyme, not Alzheimers), while the parent posts are about people actually having both diseases.