Reviews show the studies linking red meat to cancer are poor quality
The University of Washington review rated the evidence linking unprocessed red meat to disease as weak. It did not find that red meat is safe or beneficial, and weak evidence of harm is not evidence of no harm. Health is also only part of the case for veganism, and not the main part.
A large, independent review of the research found that the studies tying unprocessed red meat to cancer and other diseases are weak and inconsistent. Most of them are observational, rely on people recalling what they ate, and can't separate meat from everything else in a typical Western diet. If the headline health warnings rest on evidence that poor, people shouldn't be told meat is dangerous as though it were settled science.
The review is real, but it does not show what it is being used to show. The study is the 2022 'Burden of Proof' analysis from the Institute for Health Metrics and Evaluation at the University of Washington. It rated the evidence linking unprocessed red meat to colorectal cancer, breast cancer, heart disease and type 2 diabetes as weak, one or two stars on its scale. The reason was the wide spread between studies. It still found associations pointing towards higher risk. What it could not do was rule out that the true effect is small. It found nothing suggesting red meat lowers risk.
That distinction matters. 'We cannot be confident how large the harm is' is a different claim from 'there is no harm'. The second does not follow from the first. Uncertainty cuts both ways, and nothing in the review supports moving towards more meat. The review was also about unprocessed red meat. Processed meat is classified by the World Health Organization's cancer agency as a known cause of colorectal cancer, and the evidence there is considerably stronger.
Nutrition studies do have real limits. Long trials of whole diets are rarely possible, so most evidence is observational. The usual answer is to weigh many lines of evidence together, not to discard one side's studies and keep your own anecdotes. The many lines here include substitution analyses, cohorts of long-term vegetarians and vegans, and trials on blood lipids. Taken together they point fairly consistently towards benefit from swapping some meat for plant protein.
Finally, even if the health question were a draw, the main case for veganism would be untouched. That case is about the animals. Showing that meat may not be very bad for the eater does not show that it is necessary, and necessity is the question.
Treating the absence of proof against a claim as proof of it, or the absence of proof for a claim as proof against it.
How it applies here: The review found the evidence of harm from unprocessed red meat was weak, not that red meat was shown to be harmless. The arguer treats the lack of strong proof of harm as though it settled the question in meat's favour.