There never was a distinction between heresies in Catholicism. A heresy is a heresy. Heresy is any provocative belief or theory that is strongly at variance with established beliefs or customs. Catholicism is centralized in ways most other major religions are not. It's hard to wrap our heads around how pervasive religion was at the time of your examples but no powerful ruler, let alone the pope, would allow anyone to challenge their authority so directly. The pope was the supreme arbiter of religious matters, it was a matter of principle.

One way your follow up and your question has trouble sticking to what really happened is that it somewhat implies that in both cases it was a single pope that struggled with the monothelitism or filioque heresies. It wasn't, and different popes took different approaches, some tried conciliation to keep control as you suggest, others chose direct confrontation.

Those two heresies also may not have threatened the position of the papacy, I contend they did, but they did not take place in a bubble. The church at the time of the monothelitism crisis, for example, was struggling with other important doctrinal questions. Honorius I, under whom the crisis would be born, had been busy converting large swaths of England and turning Irish Easter celebrations into something more recognizable in the rest of catholic lands. Honorius was a monothelite and was tossed by future popes for his beliefs. His successor didn't do much of substance (mostly because he did not live long, he was pope for under 3 months) but began the line of popes that would condemn monothelitism. John IV had to contend with two major heresies, pelagianism as well as monothelitism. The next popes will be no different than the three I've named. They'll all contend with more than just monothelitism.

The early church had to stamp out any number of heresies on any given date, some benign, some major. The reason why monothelitism, even if it does not directly challenge the popes powers does challenge his authority in ecclesiastical matters and if his authority isn't ultimate, therein is created room for more serious heretical theory.

As for filioque, that one was a major threat to the power and position of the pope. Filioque is similar to Arianism in that it denies the doctrine of consubstantiality. The church moved because they saw the threat of renewal for one of the greatest heretical movements the church had seen up to this point. It became directly about the pope's right to judge what was and wasn't heretical or acceptable dogma because the Eastern church rejected the authority of the pope to change the Niceno–Constantinopolitan Creed.

Sources:

-The Great Heresies by Hilaire Belloc (I really recommend this one, its spectacular)

-The Early Church by Henry Chadwick

-Western Society And Church In Middle Age by R. W. Southern

It wasn't just the papacy that was adamantly against 'heresy', so it was not alone in being stubborn and the issues it dealt with were certainly not academic. Christological controversies had created problems since before the First Ecumenical Council, causing many, many riots, massacres and persecutions, so when the papacy became involved in the monothelete debate it had been dealing with similar issues for more than two centuries - it would be stranger if it ignored the heresy. The monothelete doctrine was also a way to bridge the gap between Chalcedonian beliefs (based fundamentally on the doctrine agreed at the Council of Chalcedon) and miaphysite beliefs (which can generally be characterised as anti-Chalcedonian), which was arguably the doctrine followed by the majority of the population in Syria and Egypt, so it was actually a major political issue. From the pope's perspective, successive emperors were literally giving in to heretics! Western churches were always suspicious of Eastern Roman attempts at reconciliation with 'heretics', so it isn't weird at all for various popes to make a theological stand.

You can see this from a cynical perspective too, as the papacy's response to the controversy can be an example of the papacy attacking the right of the emperor to intervene in doctrinal matters in order to secure its ecclesiastical supremacy. There were also revolts in Africa and Italy around the same time, some perhaps motivated by their Chalcedonian beliefs, some by more mundane political reasons, so several popes may well have wanted to use them to fuel their own ambitions.

I did a lot of reading on monotheletism for my undergraduate thesis, so feel free to ask me more questions!