Like other project deadlines, most lawyers don’t do anything until they are on deadline. They may sit on things until they are forced to act, hence filing close to statute deadline.
I get requests regularly that need to be handled ASAP. I’ll do an IME, for example, and not hear anything for 18 months, then an atty asks for an addendum or update to review “new” info within 2 weeks of a trial.
Seems like human nature. I can’t think of any advantage to waiting until the end, as cases get dragged on regardless. It’s not like waiting compresses the time to closing the case.