In the original version, why doesn't Mumm-Ra get more proactive about dealing with the Thundercats, instead of lurking inside his pyramid all the time? It turns out that anyone can steal/use the powers of the Ancient Spirits of Evil if they gain access to the appropriate chamber. Mumm-Ra doesn't dare be away from the pyramid any more than absolutely necessary.
Beyond that there's the fact that Mumm-Ra is eternal. He can easily triumph over the Thundercats simply by waiting for time to take its toll on them, letting them grow old and die while he keeps on ticking. He knows this full well, he's just coming out to battle them because he gets bored and wants some entertainment from time to time.
We also see evidence that Mumm-Ra's own plots don't quite match with those of the Ancient Spirits of Evil. At times, he both seeks alternate sources of power and sabotages other villains who draw upon them. Given how intelligent Mumm-Ra is, it stands to reason he spends more time planning than acting, and is willing to delegate to the Mutants.
The mutants have perfected a weapon that will burn down any area they choose. Why bother waiting till sundown when it's already locked and loaded on Cat's-Lair right now? Seems like a case of Villain Ball—until realizing that by not making a move till nightfall, they will not only destroy their enemies' headquarters, but also most likely immolate them all in their sleep.
There is a lot of subtext that suggests that Mumm-Ra is basically a slave to the Ancient Spirits of Evil, that he made a Deal with the Devil of sorts in return for power and eternal life, but at the same time he is Forced into Evil in a way in that he must destroy the Thundercats as they are a force for good that the Spirits don't want around, and if he fails he is tortured and if he gave up he would be destroyed and replaced. While Mumm-Ra is largely unsympathetic and seems to genuinely enjoy being evil, there are nevertheless shades of Tragic Villain and Jerkass Woobie about him if viewed from that perspective as it's unclear just how much freedom he truly enjoys. This is made explicit in the Wildstorm comics, where he was an ancient Egyptian priest named Wahankh who asked for their help in letting him take over Egypt, but when that failed they laughed at his failure and forced into eternal servitude.