Revolts weren't promoted, because it was clear how uncontrollable they could become. Many people had learned of the violent slave revolution in Haiti seventy years before and dreaded a recurrence of such horrors. And if race war began between Blacks and Whites, many unionists would leave the ranks, and take up with the other side. Though we talk about "total war," there were some things that weren't resorted to in the Civil War.
Would slaves have revolted if there'd been more energy put into agitation? Probably not, but loyalty to masters, fear of recriminations, and acceptance of existing conditions shouldn't be assumed to imply an ideological support for the Confederacy. People who live and work in a given country don't necessarily assent to the policies of its government simply because they go on living and working there.
There are different questions here that may not have the same answer. One concerns the degree to which slaves accepted their condition. The other involves whether Blacks took up arms for the Confederacy. Yet another would be what African-Americans thought about the Confederacy. Certainly numbers of slaves accepted their condition because they knew no other, feared change, or were loyal to their masters, but whether this translates into support for the Confederacy is less clear.
The way we define terms also matters. There may not have been a bloody slave insurrection, but there were plenty of small scale, even individual revolts, if leaving the plantation or disobeying orders count as acts of revolt. And surface impressions may be deceiving. Slaves wouldn't always inform masters of their true sentiments.
BTW, you may be able to find your ancestor here.