Membership start date and "member since" date not set with online contributions
This problem was seen on one of our sites and the issue was in the status rule page where Pending is selected as Member = Yes.
Pending is a reserved status that defaults to Current Member = No. Civi doesn't allow you to edit this status via UI but i see inline edit still works on these statuses. This is clearly a bug in core.

So i think it was unintentionally updated to Yes on your site since i see the same case on your attached screenshot.
Reverting this value back to No fixes the problem with the dates.