Roundup is a great product and I’d highly recommend using it to kill weeds. But “Roundup” is a trade name, the active component is Glyphosate. You can buy own brand weed killer in Tesco and many other places that contain Glyphosate for a fraction of the cost. However, it’s a translocation systemic weed killer, it’ll kill the current weeds but more will come back. I’d suggest using it as a short term measure until you come up with a better place, such as a mowing strip. (I cannot understand anyone that uses weed killer year-in-year-out on the same patch. A waste of time, money, etc.)
The Glyphosate itself breaks down in a short time when it makes contact with the soil, remaining active in the bottle and the plant only. Yes, it contains carcinogens. But alcohol, cigarettes, red meat, Vitamin C, sunlight, and many other things can cause cancer.
I wouldn’t use vinegar, eventually it’ll poison the soil. It won’t break down as Glyphosate does.
A fabric weed barrier works well for a couple of years. But natural mulch and debris, such as leaves, will collect on the surface, break down, and will soon collect weed seeds and you’ll be back to square one.
Overall, I think Thirsty’s advice is best.
Or you could use a long handler lawn sears instead of the strimmer!