Mares tail, a pernicious weed. If you aren't prepared to use a weedkiller then all you can do is to pull it all out, but that's not guaranteed to remove it all. However, there are safe weedkillers that are harmless to pets so ask in your local garden centre.

Not that Weed killer actually works on that stuff.
You actually need to bruise the stems before spraying for it to work and you would need to do it every few weeks for a long time to have any chance of success. The roots can go down a very long way. I once followed a root over three metres down and it was still descending when I got fed up. And I have seen roots on the edge of a quarry going down over 10 metres.

In my view, the only option is to take up the gravel and membrane, and replace the membrane with one of a superior quality (and possibly put down two layers). Some on the market are either not fit for purpose, or only suppress the less thuggish weeds.

Trying to use a barrier method (pond liner, thicker geotextile membrane) will make no difference - this stuff punches through anything you use. It won't come through solid concrete with paving on top, but will instead make its way to the nearest area with daylight and appear there, I'm afraid.

Blimey. It looks like the OP has a major problem, then.

Oh yes - its a reason not to buy a property, that and Japanese Knotweed (although that's far worse) as far as I'm concerned. Like bindweed, you know you're in for regular, persistent treatments and the best you can hope for is good control.

I have seen it come up through asphalt and its roots can go many feet down into the soil so it's a waste of time trying to dig it out as already suggested. Have a look at the site advertising 'Kurtail Gold'; which says it will kill marestail. Otherwise you will need to call in professional help.