Bicycling requires strength and endurance.

So does cleaning a house, but that's "women's work".

Is it that you're in control of a heavier and more powerful machine?

That's it. You didn't get it at first because made the mistake of associating manliness with things like patience, strength, hard work, endurance both of toil and hardship; all things that do make up ideals of manliness to normal people. But you need to approach it from the perspective of a wastrel, a weak, foolish, and lazy person who demands the respect and deference of being manly without putting in the hard work—something he has avoided all his life. He might praise hard work in abstract, but he has no discipline for it and doesn't respect those who actually do it, he just considers them beneath him. To such a person, the defining aspect of manliness and machismo is mastery, mastery over others and their wills, and since mastery through work is a waste of time to him, he turns to shortcuts.

From there, it's not hard to see where the thought process goes. Since strength is to him based on control and mastery, he picks something that gives him more command over the road in a direct and in-your-face way. The man who drives a lifted Ram 2500 can confront you by running you the fuck over. By contrast, in his opinion, cyclists are entitled jackasses in miniscule booty shorts who can only confront you on the road by screaming "CRITICAL MASS! FUCKING CAGER!" and throwing sparkplugs at your windows. The difference in power dynamic is proof enough to our friend of who the "real man" is.

To take the mentality to its conclusion, the easiest way to gain mastery in general is through authority, and the easiest way to get that, even easier than joining a gang, is by becoming a cop.