Yes, the Democrats were the majority party, but their constituents were made up of people who didn’t believe themselves to be Insiders.
This is not a question of who has power, it’s the question of how people see themselves.
This is exactly why people today can think of themselves as speaking “Truth to Power” when they support what are actually the default positions in our society. They see themselves as outsiders struggling against the insiders even long after they’re in control, such as in our universities.
It’s entirely a matter of perception, not of facts.
After the Civil War some people had the feeling that first class citizens were Northerners and that (white)Southerners were second class citizens, but that feeling just wasn't as strong before the Civil War.
If you were a wealthy planter, you may not have felt yourself to be aggrieved and put upon, and even Southern yeomen didn't feel themselves as outsiders, certainly not to the degree that they would after the war.
I don't doubt that Southern Democrats had feelings of alienation or marginalization during the Federalist era or during the 65 years of Republican domination that followed the Civil War. Such feelings might likely have persisted into the Jacksonian and post-New Deal eras, but the possession power in the government has a way of mitigating such discon oftents.