Edit: By “overthrow” I mean an actual attempt to topple the government, not wandering around aimlessly in a building for 4 hours stealing furniture.
If you actually oppose power you would almost certainly be FBI’d long before you could ever get to a position where you could conceivably stage a coup.
Being barred from election is a very quaint punishment for leading a failed coup.
You're highlighting the limits of "the state" as a frame of analysis. "The state" is a legal fiction that's often useful, but that fiction starts getting in the way when you look close enough to see the motivations of different factions or individuals.
Different factions and individuals use the state's tools for their pet projects all the time. This is why libs are obsessed with rules, institutions, and norms -- they want the state to function via the rule of law, not the whims of whoever is in power.