According to the timeline of events, the Balrog was trapped until 1980 Third Age (TA):
The Moria dwarves awaken Durin's Bane, a Balrog, which kills Durin VI, king of Khazad-dûm
The events of the Lord of the Rings occur primarily in 3018-3019 TA (including the fight and demise of the Balrog). So we have 1,038 years for the Balrog to struggle for power.
As best as I can tell, there is no canonical reasoning, but my guess would be that because he was initially beholden to Morgoth, he would have had to wait to build his own forces, as he would have to start from zero once he was freed. We can look at the timelines of Sauron's rises to power. Sauron was already Morgoth's chief lieutenant, and so he potentially had an easier time of raising forces.
After lying hidden and dormant for 500 years, he began revealing himself once more, and by SA (Second Age) 1000 he gathered his power and established himself in the land of Mordor in eastern Middle-earth and begun building the dreaded Dark Tower of Barad-dûr near Mount Doom. Sauron, like Morgoth, soon began raising massive armies of Orcs, Trolls, and possibly other creatures, as well as corrupting the hearts of Men with delusions of power and wealth, chiefly Easterlings and Southrons (the Haradrim).
Although Sauron knew that Men were easier to sway, he sought to bring the Elves into his service, as they were far more powerful. By about SA 1500, Sauron put on a fair visage in the Second Age.
So this first rise to power took somewhere between 500-1000 years. If we look at the time line of events again, Sauron was defeated in 1700 SA and rebuilt, but even by SA 3263 (over 1,500 years later) was not powerful enough to challenge the Numenorians. His final rise to power began in the Third Age:
In the Third Age, Sauron arose again in TA 1000, at first in a stronghold called Dol Guldur, the Hill of Sorcery, in southern Mirkwood TA 1050. There, he was disguised as a dark sorcerer known as the Necromancer, and the Elves did not realize at first that he was actually Sauron returned. The wizard Gandalf went to Dol Guldur in TA 2063 in secret to see who it was that ran Dol Guldur but Sauron, sensing that his secret identity was about to be unveiled, had fled before him and gone into the East to hide; thus began the Watchful Peace. Sauron returned in TA 2460. Gandalf the Grey stole into Dol Guldur in TA 2850 and discovered the truth. Eventually, the White Council put forth their might and drove Sauron out in TA 2941.
So Sauron's rises to power took hundreds or thousands of years, and he at some points had a baseline to start from. The Balrog had nothing to begin with, so perhaps the Moria goblins were the beginning of his rise to power. Who knows what would have happened in the fourth age if he had remained unchallenged and had his powers growing until after the elves and Gandalf had left Middle-earth?