Instead of thinking chemically, think pyrotechnically - but only if you're in an area that's not subject to wildfires. A weed torch is relatively easy to use and kills all types of weeds. Here's a photo of one in use from a supplier's web site (note that I'm not endorsing this vendor). See the text under the photo for some additional information. This particular model uses propane (IIRC, I've seen others that use LP gas).
Given that the added photo shows a fence (and car tire), then torching the weeds is not really viable for much of the area in the picture. I think using a string trimmer weekly would be a better solution. If you have dandelions, then spot-treating with a broadleaf killer (like Weed-B-Gone) would work well - you could mix up a gallon, pour it into a hand-sprayer, and spray only the offending weed.
The important point is to never let weeds set seed. If you can deadhead them, then you'll reduce the number of seeds in your own yard (you'll still get some airborne seeds and their sprouts because of your neighbors, of course). The more annual weeds that you can prevent from sprouting, then the fewer annual weeds you'll have - this cycle could make the area relatively weed-free in about five years (it's worked well in my vegetable garden - no intensive weeding, even without mulch.