I think long Covid suffers from the same problem as Lyme disease, and also whatever set of real conditions end up explaining chronic fatigue syndrome: the list of symptoms, for a layperson, sound vague enough to attract large numbers of people looking for a concrete explanation for vague but persistent feelings of malaise. Basically the things that an infomercial might invoke: are you tired? foggy? achey? have trouble sleeping? And it's a physical diagnosis with a physical cause.
I remember when I first identified what I was going through as depression and started thinking about getting therapy, someone referred me to an online chronic fatigue syndrome community. I found that it was shockingly rife with open bias against the concept of mental illness. Tons of people with classic depression or anxiety symptoms and no detectable physical problems were railing against doctors who suggested they see a therapist or psychiatrist. Over and over again, in slightly different words, I saw them express variations on the same theme: a mental health diagnosis means you are malingering, stupid, narcissistic, or out to control others by being a burden on them. In their framing, a doctor who suggested seeing a mental health professional was guilty of maliciously lashing out at a patient whose condition threatened their facade of professional omniscience. It was my personal real-world introduction to this kind of stigma, and to this day, I still think about that CFS forum when I look around and wonder what people think about me.
Of course there were kernels of truth there, that some doctors harbor stigma against mental illness themselves, and that such doctors are more likely to suggest mental health treatment when they don't trust or respect a patient, often due to bias. Some of the people on that CFS forum had stories that were pretty damning of their doctors. But many felt that merely raising the topic of mental health was proof of a doctor's bad faith.
Later, when my father was diagnosed with Lyme disease, I read up on it online and noticed the same crowd in the Lyme disease "community." Time had passed, and the stigma was not so openly expressed, but there was the same subset of people with symptoms completely consistent with depression, who were devastated by the suggestion to see a mental health professional and were determined to leave no stone unturned in their search for a more acceptable explanation of what they were experiencing. And of course there were those who saw through the wannabes and leaped to the conclusion that the whole thing was made-up.
With Lyme disease, I felt the sting from both sides: people on one side implying that my depression was just laziness hiding behind a medical facade, and people on the other side denying the reality of a disease my father had. It reminded me that some percentage of people claiming to have CFS had real undiagnosed physical ailments and were probably mortified by their unwilling association with the circus of offensive and/or nutty people who claimed to "advocate" for them. My father certainly had nothing to do with the Lyme conspiracists that demanded a congressional investigation into the theory that the U.S. government invented it as a bioweapon[0], but they're always going to be connected in people's minds.
Luckily, in the case of long Covid, medical science had a head-start. The possibility of long Covid was raised by doctors before it even happened, based on experience with other viruses, and the initial reporting was driven by the medical community as a whole. (The way many people first heard of CFS was thanks to the media savvy of quack doctors selling therapies to desperate people.)
Anyway, that's a longwinded way to say that long Covid will attract its share of self-diagnosing crackpots, but the public perception will hopefully not be affected as much as in the case of CFS or Lyme disease.
[0] It happened in early 2019: https://www.vice.com/en/article/neaxdq/the-conspiracy-theory...