Logically, it's impossible to prove a negative.
Claims by these fringe you describe state that time travel is real. Dissenters state that it is not.
Since the negative can't be proven, it is incumbent on the positive claimant to prove the truth of his claim.
This is where Titor's "2% divergence" came from. This is where these "vortexes" that are required for the HDR came from. This is where the claim that the mere presence of skepticism prevents mind reading, telekinesis, spirit communication, spoon bending, etc. came from.
All of those (and more) arise from the complete inability of these claimants to provide any evidence that their claims are true.
99% of the times I bother to even look into a time travel claim, I find quite quickly that the claimant doesn't even understand the implications of their own claim. Within 5 minutes (usually,) I can tell you exactly how they have already been proven to be utterly wrong in their specific claim.
But that doesn't "disprove" time travel. In fact, one-way time travel into the future is confirmed, well-known and fully explained by science. Nothing in science precludes traveling into the past either, except engineering.
That, however, doesn't justify the silly ignorance on display regarding this subject.
Great reply @Harte .