But even if one civilization decides to broadcast and/or colonize the
galaxy, are they really going to be able or want to contact and/or
settle every star system in the Milky Way? And out of 400 billion
suns the Sol system happens to be on their list?

If I was going to colonize another star system, I would look for ones
that are NOT already occupied so I can have all the resources for
myself. I am sure there are plenty of uninhabited systems in the galaxy,
or at least systems where the so-called intelligent species are in no
position to detect and/or do anything about visitors.

And here is another factor to throw into the Fermi Paradox - it should
be popular here, it's quite pessimistic regarding the nature of intelligence.

From the SF novel The Killing Star by Charles Pelligrino and George Zebrowski:

1. THEIR SURVIVAL WILL BE MORE IMPORTANT THAN OUR SURVIVAL.

If an alien species has to choose between them and us, they won't choose us.
It is difficult to imagine a contrary case; species don't survive by being self-sacrificing.

2. WIMPS DON'T BECOME TOP DOGS.

No species makes it to the top by being passive. The species in charge of any
given planet will be highly intelligent, alert, aggressive, and ruthless when necessary.

3. THEY WILL ASSUME THAT THE FIRST TWO LAWS APPLY TO US.

The full, pessimistic details and then some here:

http://www.projectrho.com/rocket/rocket3aa.html

Just like with the discovery of giant exoworlds orbiting their suns in a
matter of days, I am sure humanity will do a collective "Whaaa?!" when
we do discover what kind of life and minds are out there - if we can even
recognize them as such at all at this stage of our development.