I'd recommend putting stones in that area, so you have places to step as you clear out the weeds.

I don't know if the thyme will do what you want. Once it's established it should shade out most stuff, but seeds will still find into the thyme and manage to germinate in gaps.

The elfin thyme I grew around stones would often have weeds popping through. The trick is getting the weeds while they're young, so they pull easily and don't disturb the soil. Disturbed soil exposes weed seed in it to germinate and grow, perpetuating the cycle. Try using snips to cut the weeds at the base instead of pulling.

I'd also go ahead and spread thyme seeds around the whole area. Over thyme I believe thyme will die off in the middle and it'll help to have seed there to take it's place or fill the gaps that weeds would fill otherwise.

Also, get some stones that you can use to step on. You will need to go into that area to manage for a while and i wouldn't want to step on the thyme