The left wing parties in Sweden do not like the idea of a military alliance with Turkey or the US, but this invasion might make it the least bad option to them.
On the long haul, friends of the US have a much better time than "friends" of Russia, especially those with a border to their "empire".
I don't think a lot of people like being allied to Erdogan. Turkey itself is not so much the problem, it is actually a valuable member due to location and contribution.
As I said, non-membership makes involvement of Finland and Sweden in any future wars between Russia and its neighbors more likely, not less likely.
The upside is to avoid nuclear war. Believe it or not, prior to the Russian invasion, the Ukrainian peoples wanted peace, too. Joining NATO wouldn't have done so.
The idea that disputes between countries all over the map should lead to NATO involvement is dangerous. It's astounding how so many people don't seem to understand this. Of course, if you say, "How would we react?" it's "whataboutism".
You label the invasion of Ukraine a "dispute"? It's a war of unilateral aggression involving war crimes and genocide.
The point of NATO membership of Finland and Sweden is to avoid military conflicts. Putin proved to the world that he does not abide by any norm or treaty, so not joining NATO will give those countries no security from his forces should he decide attacking them is in his best interests.
Also: If Russia tries anything with the baltic states, it will almost inevitably attack parts of Sweden and Finland just to safeguard its flanks.
There is no upside to appeasing Putin anymore.