Constantine basically created the Pacacy via various gifts and it grew into temporal power.
That's far, far more complicated than that, and the grew of pontifical power was everything but linear.
Roughly, you can identify different periods in early medieval papacy :
- Ostrogothic Papacy
Ostrogoths mainly kept a neutral stance when it came to dogmatic and pastoral roles of Roman church. However they deeply intervened in its structural functioning and elections of the pope, even if Roman church kept strong ties with Constantinople.
- Eastern Roman Papacy
At first, the pope, while keeping an honorific primacy, wasn't much different from the patriarch of Constantinople, being a relay of religious imperial policies and from the nomination dependend from the emperor, then from the exarch.
Eventually, tough, they grew more resiliant to imperial policies (maybe a by-product of Lombard advance?) up to debate with the emperor about its dogmatic choices (as Monothelism).
It created a series of conflicts with the emperor, ending with an uneasy relations with Constantiople, while the pope felt more and more independent (or tried to be so) within the Ducatus Romanus' limits.
Even during this period, tough, they were far from being simple Bishops of Rome, but Patriarchs whom moral prestige was still high even in the Easterns regions.
- Frankish Papacy
Eventually tired of the ERE and Lombard pressure, Papacy called Carolingians. Which basically put the pope within a golden cage : while giving him much prestige (more than they had so far) and definitely making Roman church the first in Western Christianity, it was made in cunjunction with the creation of an empire that was concieved not as roman, but as Christian.
Therefore, the emperor decided everything including religious matter, the pope had to obey.
It became more easy for the pope at the death of Charlemagne, and his son (that recieved a religious education) was more respectful of the papacy's own interest but the collapse of Carolingia didn't let much time for that.
The pope conserved much prestige and potential legitimacy, but was stuck in Italy when it came to exerce these. While more free on these regards now (not waiting for Carolingians to give their satisfecit), the main priority was to establish some stability, as crowning kings of Italy and emperors whom main goal was to protect them in a period of renewed raids in Mediterranean and global disorder among western Christians
- Crescentii/Tusculani Papacy
Eventually the choice and policies of papacy were more and more tied up with Roman aristocracy, which was divided among others things along pro-Frankish, pro-German, pro-Byzantines lines.
It was nicknamed the "Dark Age of the Papacy", but it may be largely cuss name given by later periods in order to justify the reforms : indeed, it was an important period to establish the autonomy and local wealth of the pontifical structure (necessary for its political independence) that grew from aristocratic fortune (much more than the contrary), while the pontifical power was indeed more limited geographically, preventing important dogmatic changes.
- Ottonian Papacy
With the rise of Ottonian dynasty, and the creation of HRE, the return to ol' good days of Emperor/Pope team had to return for that the empire being really so, and the papacy taking back its (alleged) former power over all western Christianity.
It should be stressed that there's no clear separation from the anterior period : it was commonly tought as well by the papacy that such tandem should exist and the roman aristocrats mostly worked trough this.
In short : papacy was never only the bishopry of Rome, even in Roman and Byzantine times, but it doesn't mean its power grew without issue. Its political independence and universalism didn't develloped in the same time, but often independently up to the point where one was already established, and the other seen as institutionally mandatory.
For the Patriarch of Rome to be nothing but the bishop of the city, you'd probably require a TL where Rome have little political importance, which implies at the very least a very different Roman Empire (and probably with interesting butterflies on Christianity).
If you meant having the Patriarch of Rome having "only" an important symbolical, moral, pastoral and dogmatical role BUT not being prevalent about these...
No Frankish takeover and earlier Lombard presence, would be the earliest safest PoD to me : before that, the (relative, once again) marginalisation of the pope was pretty much advanced, and having Lombards kicking more importantly their way in Italy up to efficiently cutting off most of ties would help.
To make things even easier, butterfly the rise of Islam, in order to preserve Carthage as an important roman Christian center in the western Mediterranean. It wouldn't be seen as an anti-Rome, but at least relay imperial religious conceptions instead.
Having different "national" churches, meaning led by their respective king and clergy, would help. No rise of Islam meaning maintian of the relativly powerful Visigothic clergy for exemple.
Basically, a Patriarch like the others in the Eastern Mediterranean, but with more moral and legitimizing power over the different "national"
What would Europe look like if the Pope just stayed as the Bishop of Rome, retaining the spirital power of being the "Prince of Bishops" without the temporal power that being a head of state brought.
Is it close enough of what you search for?
I assume the Crusades would have happened, but would the Reformation?
Crusades wouldn't have happened in all likeness. They were
extremely tied to two important events, that may simply be butterflied ITTL : rise of feudal nobility whom social justification was military, and growth of pontifical power trough Ottonian reforms and Peace of God movements.
As for Reformation : the very same than for Crusades. There's simply too many butterflies involved there.
