Help! My Patio is Overrun with Weeds – Any Ideas?

I've got this huge patio that was here before we moved in six years ago. For about two years, I was mostly working abroad, and then last year, I dealt with morning sickness until late in my pregnancy. During this time, the weeds really took over—mostly grass, some hefty docks, and a few other stubborn plants. The gaps between the slabs are pretty big, which lets the weeds really dig in. There's also some brick paving around three sides of the house that's having similar issues. I usually avoid using weed killers since we have hedgehogs and bees in the garden. I've tried using a salt and vinegar mix, but it takes a lot to cover the area, and everything just grows back in a few weeks. My priority is my veggie patch and a few herbs and flowers (and of course, my baby limits my gardening time!). But I keep getting remarks from our house-proud in-laws and neighbors about the state of the patio, and I'd like to be able to enjoy sitting outside without feeling guilty. I don't have a huge garden budget—maybe around £100 for a solution. Does anyone have suggestions? I've considered a few options like continuing to weed (though it takes two days to do the whole area!), digging out as much as possible and using Roundup once, or lifting the slabs to put down weed membrane. I'm also thinking about sowing some creeping thyme in the cracks. Any other ideas? Thanks in advance!