Arguments that getting flu shots can cause certain serious adverse responses like autism, Alzheimer's, etc. tend to be comparable in quality to arguments that that the earth is 5,000 years old.

While flu shots very likely cause no serious adverse responses it's a quite different case then the earth being 5,000 (or 6,000) years old.

As noted in the paper, people with GBS, allergies to gluten, allergies to eggs, etc. can have serious adverse responses to flu shots, but I would expect people to know whether or not they had these sorts of problems before they went in to get a flu shot.

There no good reason to assume that everybody has an allergy to gluten knows about it. If there are known conditions under which flu shots can have serious adverse reaction we can't be as confident that we don't miss an additional case as we are confident that the earth is older than 6000 years.

Claiming that's the same order of confidence seems to me quite a stretch.