Why does Safari no longer save flash game progress?
I just found a solution!

Go to Safari->Preferences->Privacy. For cookies and website data, check "Always allow". Refresh the page and the games save correctly. But you need to risk your privacy doing so.

Edit: It could be Kongregate changing locations of Flash game file recently causing this issue.

Edit 2: I think I found the cause of the issue. I have compared the source code of some webpages on Kongregate. On the pages where the games do not save correctly (such as Anti-Idle: The Game and Amorphous+), there is a string "konggames" in the source code. In the previous states as on Web Archive, there is no such string in the source code. To show that this is indeed the cause and not just the same string appearing everywhere on Kongregate, I checked the game Solarmax 2 and found that the game saves correctly and of now, and there is no string "konggames" in the source code.

This means that Kongregate has recently changed the host of some games to this "konggames" thing. By choosing "allow cookies from websites I visit"  or stronger settings in the preference, you effectively caused Safari and Flash unable to load save correctly because you have not visited the site "konggames".