When Rome was Christianized - The popes took on the pagan title Pontifex Maximus.

not really. Gratian gave it to the pope after he removed the altar of victory from Rome's senate building and some pagan senators protested pointing out Gratian was pontifix maximus causing Gratian to give up the title and Leo I only took up the title in 440 and by that time the roman empire was pretty fully christianized.

Christianized pagan temples, sites,

no one claims they didn't. We have imperial edicts and historians pointing out specific instances of this. But that can easily be seen in a holy war type mindset where you are dismantling the old gods and their systems and not suffering as a result (as many ancient writers did)

former gods and goddesses replaced by pantheon of saints

perhaps but this is a different type of move than the other things. It's a social science claim about co opting local traditions instead of discrete acts taken deliberately.

Was the Vatican and roman Catholicism really a continuation of the pagan Roman Empire?

completely different question. the Roman Empire became Christianized in the dominate era (post crisis of 3rd century) and there are clear historical links from Christianity to the pagan world but it doesn't seem as crassly imitative as you seem to imply. What do you really mean by that statement? Your conflating multiple types of evidence which don't point in the same direction.

This is my first post on here so bare with me here. You could indeed partially say that Christianity toke over and adapted a lot of pagan sites and even practices, rituals and feasts. Before emperor Constantine adapted Christianity and allowed it to be practice within the Roman empire and later the Edict of Thessalonica which made Christianity the state religion of the Roman empire in 380 ad Christianity was only the religion of very few people in the Roman empire. From that point Catholic church has proven to be a master to take old religions and work them into Christianity. Most of the early churches are build on old temple locations and turning old pagan festivities in Catholic religious feasts. I personally wouldn't call it a continuation cause there are some significant differences. Christianity didn't evolve out of paganism and has his origins within Juadaïsm so it didn't continue out pagan believes. Christianity did take a lot out of pagan religions in order to make it easier for the people to adapt to Christianity and if this didn't help the Catholic church did often use violence to convert people anyway. As a side note in this point there really isn't talk of the Vatican in this institution and the power of the pope around 4th and 5th century was really marginal. The pope was just the bishop of Rome and most of the religious power was still in hands of the Emperor. The power of the papacy would only slowly grow through the middle ages and would still then have its ups and downs through the ages.