Kill it with fire!!!

Don't worry about killing it all, just use a decent quality weed membrane under your path. No worries with strong weed killer getting where you don't want it that way . Note that although I don't use weed killer on principal I don't actually see that it would be an advantage here anyway.

My method would be:

1) Let it grow till mid summer, when the weeds are in full growth
2) Don't let a single weed set seed otherwise the problem will spread everywhere. You could achieve this by strimming regularly
3) Spray Glyphosate on the area
4) Wait two weeks
5) Spray Glyphosate again if anything is still alive or poking through. Make sure everything is dead.
6) Cover with a layer of compacted type-1 and sharp sand
7) Lay blockwork
8) Brush a sand and cement mix into the cracks to prevent future weeds setting seed

It's not quick but you'll have a weed free path that will last for years in good condition. The only things that can really cause problems with this method are Japanese Knotweed and bamboo.

A membrane may also work, but on a narrow area like a path things like bindweed will just send out rhizones and spread the problem laterally rather than kill it.

Whatever you do, don't use fire though. It works on shallow annuals in cracks on paths etc. But deep rooted rhizone weeds and many pernenials will just grow back.