If the wind pulled so hard that the line was perfectly straight, it wouldn't
be a catenary any longer

a chain (or nylon rode, or a piece of cord string) ALWAYS has a catenary.
ALWAYS.

to pull the line straight requires infinite effort. and to pull it "just a
little bit more" when it is already stressed with a 1,000# pull is HUGELY more
than 1000#.

That is what breaks boats loose, or at least boats without 15 or 20 feet of
stretch left in the rode.