I just pile the mulch everywhere I plant. It doesn't change the sunlight recommendations or plants. Keep the wood on the top and you won't have an issue with soil nitrogen becoming tied up. Landscapers wood chips are especially good because they include a very high ratio of green leaves and bark which include a more complete balance of plant nutrients. Research ramial wood chips if you want to read the studies.
Mesquite is actually fairly well known as a nitrogen fixer, so the best soil on the property very possibly might be where you're taking those trees out. I grew up with an old mature one in our front yard and we always had healthy grass growing in the dappled shade (and the only care our grass gets is mowing) I also remember the thorns, which is the only reason I'm not suggesting you keep them.
Mimosa is a short lived nitrogen fixing tree that gives a similar dappled shade. It also has a long blooming season that brings in pollinators like hummingbirds and
bees. I'm growing olive trees to try an get that same kind of dappled shade growing over some of my vegetable gardens. The seed pods are probably hanging off your neighbors trees right now. This is a tree that has a reputation for becoming weedy, so I wouldn't spend money buying one.
If you are having problems getting trees to grow because of the clay, you might think about planting in small mounds. That's actually the recommended strategy to keep peaches alive in this area. It doesn't need to be very high or elaborate just a few inches will improve drainage around the
roots and help stop a lot of problems before they start.
For smaller than tree sized nitrogen fixers, you're pretty much gonna want to plant legumes. That would be clovers, black medic, any kind of bean, lupines (yes, bluebonnets qualify). I keep rotating different eating beans through my vegetable gardens and leave any 'weed' that I know fixes nitrogen. At the most I'll prune it back (chop and drop) if it's physically crowding out my cropping plant. This is one of the many reasons it pays to know your weeds. I don't get much use out of knowing the edible weeds, but I use all the nitrogen fixers.
This is actually the start of planting season for bluebonnets, but the seeds can be expensive. If you want, I'd be willing to save seed at the end of next season (which would be June) I've been growing them in my front lawn for the last few years. The blue bonnets shade out all the grass in that corner until the temperatures are already in the 90's and the rains are almost gone. Right now it's the thickest grass in the yard. I'm probably going to have to give it a very short cut soon, so that the bluebonnet seeds can get enough light to germinate.
Even if you can't get domesticated animals, don't underestimate the value of wild ones. Something as simple as encouraging the wild flowers can go a long way to improving conditions where you garden. I don't think you can start encouraging pollinators or predatory insects too early. The more insects you bring in, the more birds, toads, snakes, ect. Most of the next level of predators will also hunt your squirrels,
mice, and
rabbits. Every level of this system will be adding their own small contributions of manures and nutrient rich carcasses to your soil.