This makes it sound like Sweden does something that other NATO members don't. That's not the case. Sweden supports the SDC in syria, which supports the YPG, which is connected to the PKK. And this is controversial in several of the countries that do this, including the US.
While it's true that it's Sweden applying for membership and not anyone else, the logical thing for Turkey to do here is to try to pressure the whole alliance to adopt some policy change (such as a terror clasification for YPG, and/or no more aid to SDC and connected groups).
Because Russia is still the scarier enemy. If the rest of NATO approves, what is Turkey going to do? Leave NATO and accept eventual invasion?
Which is BTW similar to what's happening to EU: they realized that some states have too much power in the admission process, so they want to "speed up" the process. Probably they will go for a "3/4" of the members have to say YES. That kind of stuff. Same old, same old.
Turkey has NATO's second-largest army, for one thing.
Because the accession process provides Turkey an opportunity to apply leverage against Sweden and Finland to change their behavior. Simply refusing to allow it would gain Turkey nothing and harm them indirectly by weakening (or failing to strengthen) an alliance that Turkey depends on for its own security.
...because neither country is supporting Erdoğan's enemies? The dictator is just posturing for his own people, he'll never actually dare to vote against every other country in NATO.
I guess both Swedish and Finnish government would both need to swallow a few frogs to comply with these requirements.
At least, they won't be able to maintain a moral high horse over Turkey any more after complying.