When my trailer winch cable began to look like it might fray, or break a few of it's stainless steel strands, I replaced it with an oversized stainless cable, two increments/sizes larger than the original, and with a factory swaged connection and a new hook.
I do use rope winch lines on small aluminum and fiberglass sailboats, as I like the soft feel and flexibility.
However, a loaded C-Dory of 4,000 to 9,000 pounds would seem to need a steel cable to survive the jerking and pulsating loading put on the cable. (Though the plastic fibers are technically more elastic!)
UV light also has a very deleterious affect on plastic fibers that it doesn't have on metal.
A flat belt of slightly elastic nylon would be OK, too, if sewn properly and inspected often. They sure spool/roll up nicely on the drum.
Good that you had the requisite safety chain!
Joe. :teeth :thup