I think it should first be noted that the eastern part of the empire and Constantinople was always more Christian than the city of Rome and the rest of the western Roman Empire. Pagans did attempt to blame Christians for the sack of Rome by Alaric and the empire‘s decline, claiming that “it was clear that the ancestral Gods and Goddesses had protected the city in the past and were punishing Romans for deserting them in favour of Christ.”

However, Christians like Saint Augustine refuted this by pointing out that Alaric was a Christian himself and he was more likely a punishment by God on the still pagan city of Rome. Christian apologists also cited how Alaric ordered that anyone who took shelter in a Church was to be spared. Alaric saw himself as a good Christian and he showed surprising restraint when he sacked Rome. He decreed churches and holy sites to be inviolable, and gave sanctuary to anyone who took refuge there. “He also told his men,” according to Orosius, one of the more straightforward chroniclers, “that as far as possible, they must refrain from shedding blood in their hunger for booty.” This was something almost unique in the ancient world and it was something Saint Augustine himself recognised, writing:

But there was something which established a new custom, something which changed the whole aspect of the scene; the savagery of the barbarians took on such an aspect of gentleness that the largest basilicas were selected and set aside to be filled with people to be spared by the enemy. No one was to be violently used there, no one snatched away. Many were to be brought there for liberation by merciful foes; none were to be taken from there into captivity even by cruel enemies. This is to be attributed to the name of Christ and the influence of Christianity. Anyone who fails to see this is blind; anyone who sees it and fails to give praise for it is thankless; anyone who tries to stop another from giving praise is a madman. Let us hope that no one with any sense will ascribe the credit for this to the brutal nature of the barbarians. Their fierce and savage minds were terrified, restrained, and miraculously controlled by him [Christ]...

You also had Roman Christians like St. Jerome in his letters rationalising Roman defeats and Gothic victories by attributing it to the Christian religion that Romans and Goths shared.

”The Goths, ruddy and yellow-haired, carry tent-churches about with their armies; and perhaps their success in fighting against us may be due to the fact that they believe in the same religion.”

Although the sack of Rome was extremely shocking at the time, it was begrudgingly admitted by the Romans that the sack was surprisingly less brutal than the one Rome had suffered under the Gallic tribe when they were still pagan. It also probably didn’t go unnoticed that cities that had remained majority pagan in the Roman Empire didn’t really fare any better than those that were not. Pro-pagan Roman historian like Zosimus in the early 6th century believed that the empire had lost divine protection by neglecting pagan Gods and ceremonies. He also claimed that the old Gods would ward off Alaric and his Goths in cities where they were still worshipped like in Athens, but of course this wasn’t true.

In fact, Athens was only spared because it had capitulated and the city that suffered the most brutal sacking by Alaric was the very pagan Greek city of Eleusis, which was totally destroyed. Eleusis had been for a thousand years one of the foremost religious centres of Greece and the city had remained pagan until the invasion of Alaric. At Eleusis, Alaric destroyed the ancient temple of Demeter, the Greek goddess of agriculture, along with everything else there. Eleusis had also been the site of eleusinian mysteries, the best known of the secret ceremonies of the Ancient Greek religion, but the last remnants of the Mysteries were wiped out in 396 AD, when Arian Christians under Alaric, destroyed and desecrated the sacred sites. There’s still some debate as to whether the total destruction of Eleusis by Alaric was anti-pagan though (see Alaric in Athens by Charalambos Bouras). Either way, the Christians saw this destruction as a sign of God’s will.

Alaric told his Goths to show reverence to churches, but no respect were shown to pagan temples. They were not always destroyed like in Eleusis, but neither were they shown any protection like churches. You can see why the claim by Roman pagans that they were being punished for not worshipping the old Gods and returning to them would provide the empire with protection wouldn’t have really held enough sway to make people ditch Christianity. After all why didn’t the Gods better protect cities like Eleusis which were predominantly pagan? Why were the pagan Gods punishing them with a bunch of heretical Christian barbarians? At the end of the day people probably weren’t convinced that a return to worshipping the old Gods would really make things any better for Rome or offer their cities with better protection by warding off barbarians as so many pagans often claimed it would.

Sources

City of God by Saint Augustine

The Fall of the Roman Empire by Arther Ferrill

Byzantium and the Decline of the Roman Empire by Walter Emil Kaegi

John Lydus and the Roman Past: Antiquarianism and Politics in the Age of Justinian by Michael Maas

Alaric in Athens by Charalambos Bouras

The Archaeology of Late Antique 'Paganism' by L Lavan and Michael Mulryan

Somewhat earlier than the time period indicated, but this question was precisely what inspired Augustine's work The City of God, which was one of the most popular works in all of Western Europe for the next thousand years, and which in many ways set the agenda for how most Western Europeans viewed history during this time. Augustine confronts head on the idea that western empire had fallen due to abandoning the old gods. His responses are as follows:

1. Rome was founded on violence and sin, and thus deserved to fall (he speaks extensively about the founding myth of Romulus and Remus, and how fratricide was therefore at the very beginning of the city). Thus, it deserved to fall, and its fall was an act of justice by God, not a result of the abandoning of the old pantheon.
2. It is the nature of empires and kingdoms to fall, and Rome is no different. There is a longstanding conflict between the Earthly City and the City of God. The Earthly City, represented by Rome, is by its very nature not lasting and will pass away, while the City of God, will stand forever. Thus, a teleology which puts Rome or any other city/civilization at the center is fundamentally flawed--all should realize that it was in some sense inevitable for the Earthly City to fail, and one's efforts are better invested in the City of God.
3. As noted admirably by u/Reading-is-good, he also points out how the Christianization of the pagans softened the blow and made the sacking of cities less violent than most would have thought. Given #1 and #2, this was going to happen anyway, and so people should rather be praising Christianity that it was less bad than it could have been instead of faulting it for the invasion.

In short, Augustine's City of God and other works of that sort sponsored a shift in interpreting history from the old narrative of a city's gods rising and falling with the city to an overarching idea that all history was the domain of and under the providential guidance of one omnipotent God. Thus, the rise and fall of any particular city or civilization was not surprising and was to be expected, while the "politics" of the next world were truly what was more important overall.

Lastly, note that by the time you are asking about (643), the majority of the educated population (and thus those who would have some sense of the difference between antiquity and the present) were heavily Christianized, if not actually monks and priests, and so the tendency to look at this through Augustine's lens or something similar was very strong.

Source:

The City of God, by St. Augustine of Hippo