I can see you're happy already, but I'm just going to add to this as it's come up in case anyone's googling.
My allotment used to be totally infested and obviously I can't do the above if I want to grow vegetables. There's very little about what to do in that case online or there wasn't when I was desperately seeking a solution.
I think two things are very important.
It's seasonal
Deal with it according to the season: If you keep digging out the roots when it is most active; from late March/April-ish to late July/early August (depending on the weather) it definitely encourages more growth (I believe the roots branch). Honestly, I dug one bed out fully 3 times in one growing season when I first had the plot which was disastrous. And, whenever I used to dig to plant potatoes in March/April they were my worst nightmare mares tail beds.
Dig it out if you must (.....and sometimes that really is the only thing you can do) outside of it's growing season in autumn or winter. Then only keep picking the tops off as it appears and I mean gently cutting the top growth, not jiggling the roots with a cultivating hoe - any disturbance and it sprouts more shoots. I do literally go round the plot snipping the mares tail with my scissors in peak season.
The second thing is RAISE the LEVEL of the SOIL
After the above, raised beds totally fox it's growth. The bed I dug out 3 times in one season I then added nearly a foot of topsoil to and it is virtually gone - I get the occasional tuft (scissors!) Other beds I have managed to raise a few inches and that's also successful, though not as good as a foot.
I did have a very tall heap of soil/weeds at the back of my plot where the person before me had over decades dug up weeds and tufts of grass and plonked them. It was at least a metre high and not a single bit of mares tail in it, which is what got me thinking....
It hardly bothers me at all now after 4 years of this approach. The first three years of digging it out wherever I saw it drove me insane and made it worse. Other dig-it-out plots are also testimony to that.
It still grows in my paths, which obviously haven't been raised, but as it does very much prefer cultivated, well-composted soil it's fairly minimal on those.
Wish I had photos, but it's never something you think to do at the time and it's too late now (Thank God). I actually didn't realise I had almost none until a new plot-holder stopped to ask me why I didn't have any. I looked around and it was true!
Hope this helps someone!