Rome was sacked by foreigners twice, once ~380 BCE, and then in 410 CE. It was taken by a number of Romans in that 800ish year period, but it wasn't constantly being walked over as you seem to suggest. It was only in the first century BCE that a Roman figured out you could use a Roman army to march on Rome, and even then Sulla didn't sack it.

As for what made it special, well functionally the infrastructure of Italy was built around it (the roads literally went to Rome); if they had decided to move that would be giving up centuries of work and leaving that open for someone else to take.

There's also the infrastructure of the city itself: the work going into draining the suburba, the civic structures... for example here is a comparison from Livy. Almost certainly fictional of course, but it demonstrates how a Roman might have thought about the civic buildings of his or her city. This is Livy's description of the Citizens of Alba Longa when Rome forcibly relocated them from Alba to Rome:

Seizing hastily what they could carry, they went out of the city, and left behind their hearths and household gods and the homes in which they had been born and brought up. Soon an unbroken line of emigrants filled the streets, and as they recognised one another the sense of their common misery led to fresh outbursts of tears. Cries of grief, especially from the women, began to make themselves heard, as they walked past the venerable temples and saw them occupied by troops, and felt that they were leaving their gods as prisoners in an enemy's hands. When the Albans had left their city the Romans levelled to the ground all the public and private edifices in every direction, and a single hour gave over to destruction and ruin the work of those four centuries during which Alba had stood.

Livy doesn't mention this, but if Alban practice followed Roman (and they were both Latin cities) their dead would have been buried there as well. Relocating like this is a sort of extermination of memory, a destruction of a self-identified community. Even if Rome had been constantly overrun as you think, there were good reasons for them to stay and try to make it work.

For Rome, Livy's first book is partly a catalog of monuments (and in fact he calls his history a monument) that remind Romans who they are, or who they are supposed to be. If those monuments are where you get your communal identity, they are not to be abandoned lightly.