No, cows do not need alfalfa or corn. They evolved to eat grass. They need grass. The rest is from intensive farming practices, partly because the USA has a culture where beef is better if the cow was so overfed that the muscle tissue is riddled with fat (it's called marbling) and partly because you can raise cattle's weight and therefore selling price more quickly if they're overfed. This drives more profits at the expense of the cows' health from eating food that they struggle with, giving them recurrent stomach issues and bacterial infections. They then solve that problem by routinely giving them low dose antibiotics, which is the perfect breeding ground for antibiotic resistant strains of bacteria, causing serious problems for human health - we're rapidly running out of antibiotics of last resort.

There are plenty of countries that raise their cows in large fields instead of in barns, eating natural grass that grows purely from rain that falls from the sky, but it's less profitable and needs protecting from American imported beef which is cheaper and more fatty.

The methane is a global problem with beef and dairy, though. Cows release a lot of greenhouse gasses however healthily or naturally they're fed.