Cleavers. Galium aparine

There is a similarity, except I have a truckload of cleavers, and those plants are much larger. I should have quoted a scale, but the starbursts here are only a quarter inch in diameter. My cleavers all have starbursts that are two inches across. Also, these have tiny purple flowers. My regular cleavers have tiny white flowers. But maybe this is a different variety of cleaver?

And no, this isn't just a baby immature cleaver. The whole stand is this size, and the stems are three or four inches long. My regular cleavers, when the stems are that size, have big leaves.

Here's a picture. On the left my "regular" cleaver. On the right, the plant in question. The stem length is the same. Well, the starburst on the left is just one inch in diameter. Also, these leaves on the plant in question are noticeably sharply pointed. The regular cleaver leaves are oval.

Sherardia arvensis.

Very interesting. Thank you! Herbaria at Oxford Plants says of Sherardia arvensis "Rings of leaf-like structures arranged in tiers along a square stem give the plant the appearance of a diminutive bedstraw (Galium). However, the small, mauve-pink, funnel-shaped flowers, clustered at the tips of the stems, separate Sherardia from Galium." Field madder, as it is otherwise known, is indeed endemic here in Texas. And the stems of galium aparine are also square! How bizarre. Except for the flower color, and the pointed leaves, the stuff is like a miniature version of cleaver.

Better picture at https://aggieturf.tamu.edu/turfgrass-weeds/field-madder/. Turns out sheraria arvensis and galium aparine are in the same family - Rubiaceae. So in some broad sense, this is a different variety of cleaver!

the roots will be a really distinct red colour (I grow sherardia for the dye).

Sherardia arvensis has been proven to be deeply imbedded in Galium and is now Galium sherardia. The people at the time wanted to name a plant after William Sherard, and they decided on Sherardia arvenis, which was the only species for the genus. Sherard was friends with Linaeus, but Linaeus never objected to the confusing naming and placement of Galium sherardia. It isnt native(endemic) to North America.

https://herbaria.plants.ox.ac.uk/bol/plants400/Profiles/ST/Sherardia