I have a Sears chain-driven garage-door opener that's about ten years old.
It's in excellent mechanical condition but lately it's very slow to begin
the opening
or closing cycle. I find myself having to push the remote button for 8-10
seconds at times
before anything happens.

The batteries are newly replaced, I've tried repositioning the pickup
antenna, but
so far nothing seems to help.

If you push the hard-wired button on the wall, does it open immediately?

If so, then the problem is related to the receiver/transmitter. Perhaps your
transmitter button is worn out. Try cleaning the switch or use a spare
transmitter.