For Frank, real power is in controlling people's motivations, not their actions. Sure, he could command someone to polish his shoes for him, but that wouldn't get him anything because they don't want to do it, they're just doing it because he told them to. What he prized in Meechum was his blind, unquestioning loyalty. Meechum's entire sense of self was wrapped up in his job, so much so that he was willing to subordinate even his sexuality to it.
Frank also knows that he has to be willing to get his hands dirty and play the part of a public servant. In the first season they showed him at a fundraising event, pouring drinks. Sure they could have hired a bartender, but that wouldn't have the same effect. Frank has carefully cultivated that image for decades, little rituals like shining his own shoes help him to stay grounded and remember where he came from while he's chasing lofty national goals. It wouldn't do for him to have a moment of weakness and actually openly display arrogance.