One problem is that people don't use it correctly. We have someone on this site who works on voting machines. The results that company was seeing were that people were only approving one candidate rather than picking everyone with whom they agreed more than they disagreed. When that happens, approval voting devolves to first-past-the-post, and obviously if we liked FPTP, we wouldn't want to change voting systems.
Another problem is that approval and range (score) voting don't necessarily pick the best outcome even when used correctly. See Wikipedia. They do not comply with either the Condorcet criterion or the later-no-harm criterion. Contrast with Instant Runoff Voting (which meets later-no-harm) or Ranked Pairs (which meets the Condorcet criterion).
Beyond all that, many third party supporters would prefer to be able to say that they wanted to support their candidate first. In approval voting, you can't indicate that you prefer your third party candidate to the major party candidate except by leaving the major party candidate off the ballot. But the whole point is that third party supporters want to support their best candidate while still expressing a preference between the two candidates with legitimate chances to win.
And of course it generally wouldn't make any difference. In most elections one of the major party candidates would win anyway. The real goal for many third party supporters is not to change how single candidate elections work but to switch to multiple candidate legislative elections. Because third parties can win seats in proportional systems like single transferable vote. And the major parties might break up into more ideologically focused factions. I.e. the Freedom Caucus and Main Street Partnership might split away from the Republican party. Same thing with the Social Democrats, the Congressional Black Caucus, and the Blue Dogs among the Democrats.
