They didn't, or if they did they end up right back at white supremacy in the end. There's quite a lot of gossip and sometimes white enslavers do manumit their enslaved children, occasionally in enough numbers to create a small "brown" class who occupy a tenuous position between white and black but that's mostly something you see in the Carolina lowcountry and New Orleans. In any event, they're no real threat to white supremacy or to the security of slavery in general both because they're more socially connected to their white relations and few in number. Free people of color are more common in the Upper South (north of the Cotton states) but there the combination of greater numbers and that many of them are freeborn thanks to the wave of emancipation and manumission toward the end of the Revolutionary era means to whites not a lot more than that their fathers and grandfathers made some regrettable mistakes and it's imperative that so far as is possible and practical any free black person must be treated the same as an enslaved one.

As to why all of this is, we need to step back a little and consider how we, and they, understood whiteness. Most enslaving states have at best hazy laws on what counts as white or black and the real practice is a combination of looks plus the white observers' and white community's tradition. They do understand a category of mixed or multiracial people, but for the most part nineteenth century Americans don't go reckoning things in terms of 1/4 black or 1/8 black and so forth. Whiteness isn't something you mix in with blackness, both in the prevailing racial mores and on a theoretical level.

What's that mean? Whiteness isn't, in itself, understood as an identity or a distinctive trait. Obviously they can look at skin color and the like and make all kinds of generalizations that are almost uniformly vile, but being white isn't having something. Instead it's a lack of distinguishing, othering traits. White is imagined as normal, the default human being: Adam and Eve were white. All the world's great civilizations were white. People only got to be other colors because something had gone wrong with them, for which there are a variety of explanations with the less awful sort envisioning blackness as a kind of reverse-leprosy that might someday be cured or otherwise the product of tropical environments and since non-white people don't have the things that whites value: civilization, Protestant Christianity, etc they are obviously deficient. You can't, in this discourse, actually be just a little white, or even really half white. It's effectively a binary: one is all white not at all. Whiteness is a kind of purity lost with even one drop of non-white blood.

As such, enslavers are generally not troubled on a theoretical level by white-appearing enslaved people. There are recurrent fears that one might accidentally treat a non-white person who can pass with some kind of minimum level of human decency or permit oneself to mix with them socially in a way that's debasing to the white person -as an equal, basically- but those are fears about imperiling one's own whiteness and consequent good standing in the white community. There's little thought that such contact can uplift any black person, since as a practical matter the character of non-whites is presumed to be fixed. It can't actually turn a white person black either, but it would be a form of scandalous behavior for the class of whites to whom it mattered most. Poorer whites, of necessity, lack that luxury. That doesn't mean they're not keen to get it when they can, though. This conviction of racial fixity persists even in the face of black achievement in arenas whites generally value, since it's not really empirical to begin with and is rather based on a commitment to the continued theft, torture, rape, and destruction of lives imagined as non-white for pleasure and profit.

Manumission rates don't have a lot to do with any of this. Enslaved people do get manumitted, in part and sometimes, because they have substantial white heritage in common with their enslaver but there's no general expectation that lighter skin increases one's odds. Per the prevailing racial mores, already well-established by 1800 and only hardening over time, no enslaved person should be manumitted because the enslaved need enslaving and black Americans walking free is a dire threat to white American civilization. That those "black" people might have lighter skin than someone one knows to be "white" doesn't enter into it.

Does that mean that the law makes people enslaved more than race does? Not quite. The idea of whiteness predates laws fixing its scope and largely comes into being in the future United States alongside slavery. By the 1660s, there's a clear and hardening racial line. The exact progression is difficult to trace in the colonial Chesapeake due to limited sources and what whites mean by "race" shifts a lot over the centuries. By 1800, where you've asked, the older conceptions of race as shaped by environment and hierarchical, but in principle flexible, are in retreat. These held that blackness (and Indian-ness for that matter) might be "cured" and those who have such status imposed upon them by whites might eventually whiten up themselves...though it would take more than an individual lifespan and might take thousands of years.

In the nineteenth century we have the full flowering of biological racism, replete with anatomists measuring things like skull volume...and cheating on those measurements. Most American thinkers are leery of insisting that black people are a separate species -that would conflict with Genesis and that's just not acceptable- but they can get very close without technically crossing the line. So conceived, black people are not capable of improvement or change. Someone with white and black ancestry is possibly not absolutely black in a blood quantum sense but their very status as multiracial is proof both of their inferiority to whites and a sort of disability in itself. This all gets very quickly into grotesque talk about mules and how "amalgamation" produces people inferior to either race on their own.

To have a purely legalistic system, wherein the law enslaves regardless of color, is certainly not impossible. We have examples of non-racial slavery, after all. That's just not what happened in the American South. Only the most extreme and weirdest proslavery theorists will argue that poor whites ought to be enslaved. (Regrettably, the very novelty of the argument is why George Fitzhugh is so remembered as a proslavery thinker and his prominence makes it easy to mistake him for the norm.) If one can somehow prove that one is white, one has a legal right to one's freedom. That's not to say that whites can't treat poorer whites very badly, since they obviously do that, but there's a whole world of torture beyond what they would countenance for whites available to them and employed against black Americans. A white man might be whipped, in extremis, for some sort of crime. A black person can be whipped for any reason including the pure pleasure of inflicting pain, as a matter of routine. There's a lot more than law going into that.