I wonder if overlap in some of the slower chanties is just natural. I've heard something like with chanties like Mr Stormalong being sung just by your everyday revival group who can't have ever sung a chanty as work song, the chorus coming in quick and sharp before the lead drawl is finished.

Regarding overlap from the chantyman with the lead, remember his job was at least 3-fold:
To keep the men in time
Inspire them to great effort
Get the job done as quickly as possible, drive it along.

In the case of the last of these one can easily imagine the chantyman coming in during the final note of the refrain.

Also remember that at the height of the days of chantying ordinary seamen were practically slaves and if they didn't respond smartish to any command a rope's end would be the least of their worries. There was also competition between watches.