You're not wrong that most anitslavery whites were not especially fussed by how harmful slavery was to black Americans. Some ofthem specifically opposed it because they believed that legal slavery where they lived would mean they would end up having black neighbors. They did have moral arguments about how it was harmful to whites, including white enslavers, but they're almost nothing at all like what you'd first think of concerning slavery's immorality.

That said, you did have a question and I came here to answer it. :) No kind of national campaign for better treatment of slaves was ever tried in the US and one is unlikely to work because it doesn't answer any of the major objections about slavery nor wold it be likely to entail a regime of laws that enslavers would find remotely acceptable, doubly so if they're imposed by people of known hostility to the institution. So let's take those in turn.

Essentially no slave wants to be enslaved, so to keep them that way enslavers rely on a brutal system of terror, torture, violence, and rape. Making that better and still having slavery is hard to imagine even on a conceptual level, but Southern whites do discuss various melioration schemes and pass laws that set some kind of legal minimum standard of treatment for the enslaved. Beyond that there are informal efforts by white moralists to lay advocate a kinder, gentler slavery that is somehow still slavery.

The problem with these laws is that the enslavers write them and enforce them, which means they're written to allow a lot of brutality and enforced so as to permit more. Enslavers from the get-go are extremely hostile to any effort by the state to insert itself into the master-slave relationship, viewing that as an encroachment upon their sacred property rights. They view those rights as the wellspring of all liberty and attempts to curtail them as inherently tending toward abolition. Even if they don't do that, then they tell the enslaved that their are limits to the power of the enslaver, which encourages disobedience by breaking the facade of white unity. That, enslavers believe, will in turn inspire the enslaved to rise up and give unto the enslavers what they so richly deserve, whether by means of poisoning and other treachery or outright violence. The enslaving class understand that as an immediate prelude to a genocidal race war.

Even actual voluntary efforts are difficult to sustain for the same reasons. Christianizing the enslaved becomes a big priority in the nineteenth century, generally understood as both for the good of the enslaved and the enslaver. They think that saved souls will make better, more dutiful laborers. Many enslavers take on the job personally, sometimes in defiance of laws against teaching the enslaved to read. Special edited Bibles are produced for use by the enslaved too. But go too far with this and your peers start to think you're soft on slavery and undermining the whole system, wittingly or otherwise.

Let me give you an idea. After the Denmark Vesey conspiracy, which may or may not have been a real plan for Charleston's enslaved people to rise up and take the city, South Carolina embarked on a debate about the education of slaves. Vesey's conspiracy, if it was a real thing and not the product of white paranoia, hatched in part out of the African Methodist Episcopal Church he helped found in Charleston. (This is the same institution that was the scene of white terrorism more recently.) A black church, even if it had some white clergy and minders, meant black Americans getting together. They might talk and give each other ideas. White South Carolina wasn't about to risk that again. Teaching slaves to read drew serious opprobrium and even oral religious instruction by whites came under sustained attack.

Case in point: in 1831, an enslaver named Francis Goulding established a Sunday school for the enslaved. They had to have written permission from their enslavers to attend. He himself enslaved people, so he should be reliable. He also opened his school to any white who wanted to watch and make sure he wasn't crossing any lines. Goulding used pictures to illustrate Bible passages and got into trouble for it, because white kids learned to read with the help of picture books. He switched to writing out passages for the enslaved to take to their enslavers, who would read the passages to them for memorization. That also taught the enslaved how to read, so it was likewise deemed no good. Finally he was down to strictly oral instruction and no homework at all. Even after that, he still got called before the city council frequently on suspicion that he was undermining slavery.

If pretty religious class in a pretty religious era isn't willing to let one of their own teach a brand of Christianity to the enslaved which is meant to make them more obedient slaves, they are certainly not going to take kindly to the entreaties of meddling, local divines that they cut back on whipping, preserve enslaved families, etc. Indeed, they deem the lash an essential ingredient of slave management to the point where if you don't want to or can't bring yourself to lay it on personally, hiring people to do so is a normal thing. Lax enslavers and overseers are marked often marked specifically by being a light touch with the whip. The space for a genuine, durable melioration campaign doesn't really exist.

Nor would such a campaign answer antislavery arguments well at all. Antislavery whites are not that concerned with the morality of slavery insofar as it relates to the treatment of black Americans. To the degree they have a moral argument against slavery, it's about the superiority of free labor and a desire to maintain their village-centered agricultural system of smallholders. They believe it a better system, but also that slavery would constitute intensely unfair competition and limit opportunities for poorer whites to advance themselves, as rich enslavers will buy up all the land and used enslaved people to work it. Furthermore, the presence of large numbers of black Americans is extremely odious to many antislavery whites, to the point where they pass laws banning any black American, enslaved or free, from whole states. These laws are hard to enforce, and harder still when they try to remove the small numbers of free blacks already present, but they are expressions of sincere and deeply felt animus.

On a more sophisticated level, antislavery whites agree with one of the nation's most famous proslavery thinkers, Thomas Jefferson, that

There must doubtless be an unhappy influence on the manners of our people produced by the existence of slavery among us. The whole commerce between master and slave is a perpetual exercise of the most boisterous passions, the most unremitting despotism on the one part, and degrading submissions on the other. Our children see this, and learn to imitate it; for man is an imitative animal. This quality is the germ of all education in him. From his cradle to his grave he is learning to do what he sees others do. If a parent could find no motive either in his philanthropy or his self-love, for restraining the intemperance of passion towards his slave, it should always be a sufficient one that his child is present. But generally it is not sufficient. The parent storms, the child looks on, catches the lineaments of wrath, puts on the same airs in the circle of smaller slaves, gives a loose to his worst of passions, and thus nursed, educated, and daily exercised in tyranny, cannot but be stamped by it with odious peculiarities.

In other words, in a slave society, enslaving is the most central model for the exercise of power and control. Whites who learn to dominate slaves through violence and exert unlimited power over them will naturally try to use that same power upon free white men. Despotism breeds itself and is fundamentally not entirely compatible with the type of freedom to which antislavery whites aspire, at least in principle. (These same whites don't mind much at all when Native Americans are treated with genocidal brutality and don't fret much about the fell influence of slavery on politics as long as it stays far away from them personally.) Treating enslaved people better doesn't really fix that, because they're still enslaved,

So basically, any form of amelioration that's compulsory on enslavers is dead on arrival because they will not accept it. Then on the other side even if they did accept compulsory good treatment standards it doesn't resolve the chief issue for antislavery whites in opposing slavery in the first place. Nor is it likely to appease aboltionists, who are concerned about what slavery does to the enslaved but understand that the horrors are inherent in the system. There's really no getting out of it, which is part of why the various "compromises" before the Civil War amounted to slavery receiving new capitulations from the free states and giving up nothing or almost nothing of material value. The white South has a non-negotiable, absolute commitment to slavery at least from about 1800 onward and probably from a century or so prior. The white North does not have a similar commitment to antislavery even in 1860, but has just barely come around to the idea that containing it is of controlling interest.

Sources

Much of this is standard survey stuff. The classic is Potter's The Impending Crisis, which hasn't aged so well. Levine's Half Slave and Half Free and Varon's Disunion! are more modern treatments. For the southern commitment to slavery, Cooper's Liberty & Slavery is essential. The fate of Goulding's Sunday school is from Freehling's Prelude to Civil War, a classic, probing study of the Nullification Crisis.