I had a vegetable garden overrun with witch grass once. I didn't know what it was at the time and used a rototiller on it. Big mistake. Made it much worse. It got to the point that I gave up on the garden for two years. Then after research, I decided to use a large piece of clear plastic leftover from an ice rink we had n the yard, and covered the entire area that was about 25x25 feet, with the plastic, weighed it down with bricks and left it there for a whole year. It worked great. We were able to reclaim that area and I got rid of about 90% of the witch grass. I don't know anything about bermuda grass, but I'd research ways of getting rid of that. I can't see using a chemical to get rid of it, in a 50x60ft area where you are going to grow food. If you can't do that, I would think thick cardboard with heavy mulch would be your next best bet. And it will improve the soil and increase your earthworms.

Your links: 2 Those look like they might be 2-3 years old and, being trees, will be hard to remove. Your options are mechanical or chemical. Using chemicals would leave you with stumps. Baby trees are a lot easier to deal with when they first sprout, so be on the lookout for new ones and pull ASAP before they become hard to handle.

I agree with mulching. I worry about extending my garden beds due to the wild violets. We managed to pull the bulk out of the front bed, but it took an entire summer. My in-laws were visiting and bored during the day so would go out and dig down to the roots. They are persistent buggers! I have followed up this year by putting a second layer of thick mulch and taking any foliage off in order to starve the roots. Beside the stray leaf and weeding a lot before the mulch was put down, I have barely needed to weed. Whatever does grow through is weakened by having to push through the mulch. My lawn is almost 100% weeds dispite reseeding every Spring and Fall. I am planning on reseeding the lawn soon for the fall and then placing the feed pellets as an organic preemergent in the Spring. Might help with the garden, too if the lawn stops being one huge weed.... :)

Before you invest more money for bigger solutions, consider how much time and money you have to actually be handling this. A good lawn is not just a 1-season job of doing something to make it look good. You may have to do something for 2-5 years before you see the result you want. Laying down sod for the entire yard is a fast option but if you aren't taking care of it, you just used a lot of money and will have similar issues again (just like you have described about your previous attempts) A non-grassy solution is that maybe you can turn your entire backyard into a kind of playground or xeriscape area for your kids. Rocks, mulch, pea pebbles, etc -- they will allow for your kids to be able to play in the yard, and for you to be able to spot unwanted plants quickly. Where you do want a particular group of plants, you can do a small bed, either in-ground or in a raised bed. But even at that, it's still something you'll have to maintain throughout the year, because unwanted plants have a way of blowing in through the wind, animals, or even popping up through landscape fabric. If you do have a rototiller already, it's still a good idea to use it to turn the weeds under if you don't want to pull them out first. After that, maintain the area by actively pulling out weeds. You can even make it a game with your kids. Even when I hand-pull weeds, the area I'm working in does get better without high cost of money, but it needs my continuous vigilance to keep it that way.