Mine's about 9 years old with mild usage but it seems to be getting stretchy.

would it really matter, as this line does not hold the sail once it's hooked on at the top?

I guess if it's stretchy enough it might be hard to hook/unhook... Have you tried reversing the line? I did that once when my main halyard started to fray near the top

My concern is that it might break while raising the sail or lowering it which would necessitate moving the boat to another area tipping the boat and then rerouting the rope.

I have reversed the line recently.