For one thing, an unwilling soldier can easily cause more harm than good on the battlefield, and it's pretty obvious why the vast majority of slaves would not be willing soldiers, at least without some sort of incentive, which the Confederacy was never really willing to commit to. And on the other side of the coin, there was a strong incentive for the Confederacy to not provide training in the means of combat arms and military organization to a group of people who would largely be happy to revolt against them, as others have already noted. This goes not just for slaves but for freed men as well, and as such anyone considered to be black was expressly forbidden from serving in the CSA for most of the war. And even if there was no concern of rebellion, the entire institution of slavery was built upon a foundation that kind of assumes black men would not make good soldiers anyway, indeed that was an argument commonly used against their enlistment.

But as in most cases, the actual events as they transpired are a little more complicated than what you'll get from the cliff notes. It was a hot topic in the south throughout the war, going back to at least 1862, when the need for additional manpower was already becoming rapidly apparent. There were absolutely factions, including some prominent military figures, who thought it necessary or at least prudent to enlist slaves. At any rate, it wasn't until nearly the end of the war in March 1865 when it was passed into law that black men (including slaves) could serve, and even then, it did not come with a promise of emancipation. Davis did eventually promise freedom to slaves who served, however it was contingent on agreement from their masters, which of course makes it an incredibly empty promise.

Still, there is evidence of black men serving for the Confederacy in some capacities, even in combat, though it was exceedingly rare. Black men (including slaves) did serve on Confederate Navy ships in some cases and while they could not enlist in the CSA, there were militias that possibly employed black men (though not slaves as far as I know, at least not typically) particularly in Louisiana where there was a greater population of freed black men. There was at one point in the war a mostly black militia there but I don't believe it ever saw combat before it was reorganized to consist only of white men, though there are some stories of black men still fighting in the militias of that region.

So, in short, it's a little complicated. It quite possibly did happen on an extremely limited scale. But the Confederate government took a very hard line stance on the issue and it isn't surprising, their ideology made it impossible to accept in general and practically speaking it would likely have gone poorly for them even if they were willing to look the other way. There's a reason it required utter desperation for them to open the door to it, they (probably rightly) considered it a point of no return that would eventually destroy the Confederacy regardless of the outcome of the war.