For the most part, red meat has only been really bad for you since the end of World War II. A glut of manufacturing capacity and petroleum lead to a massive increase in the mechanization of grain farming (especially corn) in North America. This led to a precipitous drop in price due to over-supply, and farmers turned to feeding it to livestock to produce a "value-add" via conversion to more-valuable animal protein.
The problem is that the digestive systems and metabolisms of grazing animals are suited to forage diets (grass), not grain. Grain has much higher ratios of Omega-6 fatty acids compared to the mostly Omega-3 fatty acids found in the non-grain parts of grasses, so feeding grains to grazing animals greatly elevates the Omega-6 fatty acids that then end up in the meat (and have other health consequences for the grazing animals). Eating the meat from animals fed this way ends up having health consequences for humans.
I would recommend meats such as pastured poultry and grass-fed, grass-finished beef (in addition to fish) to reduce the Omega-6 fatty acids compared to the typical American diet.