Going by this explanation, the expression "an axe to grind" has nothing to do with grinding or axes. Thus, options A and C areincorrect.Option B is correct as the idiom means to have a strong opinion about something that you want people to accept and that is the reason why you do something. For example, "Unlike Trump's claims, environmentalists have no political axe to grind". It means they do not care for politics, they just want to save the planet.
Options D incorrect as the idiom has no history of being used in such a context.