Sorry to be the bearer of sad tidings, but this is really hard to control in a situation like this - you are NOT going to get the root out from a tiny crack in hardscaping. You will have to keep removing the top. If you don't want to use chems, either use a weed hoe/tool to reach for you, or hire someone.

Alternately you could cover with something opaque like cardboard for a few weeks.

Spurge is my weed of the year. Hurray for spurge!!! Apparently, last year's drought caused my neighbor's lawn to degrade to an even lower state than a mix of pure crab, goose, and dallis grass. That lowest state is apparently 'spurge lawn'. This is what happened when you're mentally defective enough to cut on your lawn mower's very lowest setting, especially during a drought.* It doesn't blow over as easily as those grassy weeds, thank FSM, but still, all my beds and sections of my lawn are filling with the stuff, whereas in any prior year I would have called it a very minor weed. Most selectives will get it...fortunately it isn't some hell weed like cleavers that can only be controlled with products like 'Change Up' that are pricey.

* turn it up to 11! Yeehaw!

Don't get me going on crab grass, ribwort and dandelions! They are getting to be a bigger nuisance that the spurge. It is almost impossible to pull out crabgrass growing in a narrow crack!

For weeds like spurge growing in the cracks of paving, I'd have no issue using whatever herbicide was necessary. I'd probably start with horticultural vinegar for any annuals like the spurge but I'd certainly consider something more heavy-duty for deep rooted perennials.

I have a completely different philosophy regarding weeds in planted areas. Those I prefer to remove manually whenever possible.

A bit late to this, but you can pour boiling water on it - just wear closed toe shoes and gloves in case of splashes.

You still need to remove the dead plants later tho.

I also use my electric weed trimmer on such things.