Welcome to TBN, For God and Texas, you are close to my world, I'm just North of Georgetown. Your experience matches mine regarding mesquite eradication... and, I've almost completely eradicated mine after quite a few years of spraying individual plants that come up from seeds each year.
OP, the answer to your question about cutting mesquite and then covering the stumps is as follows:
Mesquite is a legume... and it has buds that run along the roots for quite a ways underground... and the roots not only go down, but sprangle out underground... unless you completely soak the ground to a depth of about 1 foot and out beyond the drip line of the plant, you will not get all these buds via the cut and then spray approach. Unless you get EVERY one of the buds, the plant will regrow. This approach is neither effective or cost effective.
Either reclaim or remedy, properly applied, singly or in combination, will kill the plant...80 to 95% of the plants anyway....the herbicide enters the plant via the leaves (and then to the roots) or the stem (then the roots) depending on the herbicide. New plants will continue to sprout from seeds already in the ground for at least 15 years .
The point is, mesquite is NOT like other woody plants you have experience with... and, you simply are doomed to failure unless you follow the best recommendations of those who have studied it and have gained years of experience fighting it.