Will the plant naturally send up offshots that will
more broadly cover the trellis, or so I need to do
something to encourage it to do so, like nip off the
top of the tendril?

IME (which, I will admit, only was 3 years of growing
them) it was 'one seed, one plant, one vine.' I don't
recall seeing offshoots (more than one tendril) from
a single seed (but, then again, I didn't watch them
all that closely) and I never attempted pruning the
leading portion to attempt to make it branch. The way
I got the morning glories to cover the entire area I
wanted covered was to plant a bunch of seeds then thin
them out if they needed to be.

Tracey