Balrogs were servants of Morgoth, Sauron was a Lieutenant of Morgoth, Sauron commanded Balrogs under Morgoth, Sauron became head evil overlord after Morgoth's defeat. Sauron gathered all evil armies to himself in order to cover the world in darkness. Why no mighty Balrogs in the army? By most accounts there were still several lurking about in dark places.
They were scared as hell. Hiding away was what they were doing.
some few that fled and hid themselves in caverns inaccessible at the roots of the earth; [The Voyage of Eärendil]
I do not think unless Melkor himself came back they'd have had the stones to show themselves. They, like Sauron, were Maiar.
some great, as Sauron, or less so, as Balrogs. [Myths Transformed; Orcs]
They were peers but that does not mean they could not work with him, even as a subordinate in some way, such as Saruman.
Balrogs were Maia just as Sauron was.
Morgoth was a Valar, a more powerful being than a Maia.
Maybe Balrogs were happy to take orders from another Maia whilst both served under a Valar but once that Valar was no longer in charge they probably wouldnt want to take orders from someone they probably see as at most an equal.