My guess is some sort of Ilex but I don’t know much about them or what kinds you’d see in Texas. If you look at your pictures there are actually black berries visible in all of them, particularly in pic. 2.

Lol, DOH! It was dusk and I was in a rush...and didn't even notice those!

So, I went back and took some photos of its dark, ~3/8" berries and seed inside...

The seed itself is slightly curved, wrinkled, about 3/4 the size of the whole berry, and seems to resemble that of a Japanese Privet?

Opposite leaf arrangement would rule out Ilex. Privet is a possibility. Is this evergreen, or at least tardily deciduous?

Don't know about the seeds one way or the other but otherwise look at Forestiera. (There being not much flesh on the fruit is in fact one characteristic of that genus. But otherwise there are other features to this plant that make F. seem a candidate).

It kinda reminds me of my inkberry holly (Ilex glabra), but I'm probably wrong.

Ilex have been eliminated from enquiries due to opposite leaf arrangement.

Forestiera are known as native privets. There are 8 species that are more environmentally friendly than the non native Chinese privets that are destroying vast areas of southeastern woodlands. Our native flora can't compete with the Asian thugs. I think Forestiera is way more beautiful too.

^ Exactly! Which is why I'm trying to positively ID these. Are they super-invasive Chinese Privets...or a native? Should I be helping to remove them, or just completely leave them alone?

Well, if each berry contains a single seed, you can be sure they’re not any kind of privet.

^ You sure? Hmm, I'll have to check that on some Japanese Privets...?

If not, what about invasive Ligustrum quihoui (Waxyleaf Privet)?

Apparently privet fruits can hold between 1 and 4 seeds, so a berry having just 1 seed doesn't rule Ligustrum out. I'll have to find out how many seeds Forestiera fruits contain. So I looked at all your native Forestiera species, as you probably did, and the leaves were no match, there are sime similarities. I was thinking L. japonicum was a match, with the pubescent stem, but my botanist friend in Texas has confirmed it to be Ligustrum quihoui. If you'd the explanation well japonicum has much longer, wider leaves, most often with entirely glabrous stems. The same is true for L. lucidum, L. quihoui, and L. sinense have shorter, thinner leaves, and usually pubescent stems. I've noticed the pubescent stem character to be less consistent with L. quihoui and L. sinense than the glabrous character with L. japonicum and L. lucidum.🤯🤣 what?

I stand corrected on the number of seeds.