What exactly makes red meat (probably) unhealthy?
The gut microbiome is partly to blame, at least for the associations of red meat consumption with heart disease.  Red meat has a high concentration of L-carnitine, which is metabolized by bacteria in the gut to produce trimethylamine, which is further metabolized by FMO3 in the liver to produce trimethylamine-N-oxide (TMAO). High blood levels of TMAO are associated with higher risk for both cardiovascular disease and early death from any cause.
Sources on the relationship between the gut microbiome, L-carnitine, and heart health:
New study links L-carnitine in red meat to heart disease
Red meat, TMAO, and your heart

Or is it possible that there is some deep genetic reason why the flesh of mammals is more harmful than the flesh of birds and fish for human consumption?

The NIH gives the following values for the carnitine content of different foods:




Food
Milligrams (mg)




Beef steak, cooked, 4 ounces
56–162


Ground beef, cooked, 4 ounces
87–99


Milk, whole, 1 cup
8


Codfish, cooked, 4 ounces
4–7


Chicken breast, cooked, 4 ounces
3–5


Ice cream, ½ cup
3


Cheese, cheddar, 2 ounces
2


Whole–wheat bread, 2 slices
0.2


Asparagus, cooked, ½ cup
0.1