What is this weed and how to eliminate? Also, a bonus!

Bermuda, Zone 8A.

Some background: When we moved in this house a few years ago the back half of the yard was a wasteland. There were shrubs/trees/weeds EVERYWHERE to the point it was uninhabitable and couldn't even walk in it. 11 months ago in February of 2023 we took out the entire yard and sodded. Apparently the soil in that half of the yard is just utter shite because we're still seeing some issues with problem weeds and brush wanting to come up.

I've used all the basic Ortho/Scotts/etc regular weed killers which do help, and late last summer got some Celsius WG which I used which has also helped. But once the weather started getting cooler in the fall, I've got these little bastards coming up. Apparently all the of the above didn't knock it out. So to get to question int eh OP- what is it, and how do I permanently resolve? I did recently buy some Lesco Stonewall with prodiamine that I will put down once the soil temps warm up in a few months but not sure if that will target whatever this is.











BONUS:

On the topic of brush/shrubbery, these are the main nuisance that still sprout up. I either cut them with mowing when they're still tiny, pull them up by hand, or have occasionally just taken a paint brush and directly applied some roundup/brush killer. That last method does seem to work but there's far, far too many of them to do that to all. Any ideas on how to eradicate this shite?