How do they consider which "new" old song to test to periodically add to their individual station?
New songs:
First, stations receive "promo copies" of new releases. They can look through those, often focusing on "star artists" and newer ones who have had several recent hits.
They will get calls and messages from the record company pushing the release.
Then they may scan the data online to see what new songs have done well in streams. They will look over the next few days at what other stations they respect are doing with the latest releases.
They look at format-specific programmer / music websites to see if there is a "buzz" on any new song.
At any time they may decide to add the song, or wait or push it aside. Often, a new song by a huge artist may be instantly added. One by an unknown artist may to to the "maybe -- let's watch it" group.
Old songs:
Generally, everything that charted has been tested multiple time. If one never did well, we don't test again. Otherwise, we test over and over the songs we play and ones that are borderline that a long rest may have "mellowed" and made playable again.
Usually, a long-time in-format station will look first at its own past research (corporate or local) and look for songs that might be playable. But generally, we move songs that are sustainable from current to recurrent to gold, adjusting play based on test scores.
There are not that many songs that we never played that suddenly might become playable.
The idea that old songs played in hit movies or TV shows are suddenly playable ignores the fact that the percentages of people exposed to music that way is minimal.
And then each PD and MD has their own "sense" of what might be "the right add" to give some variety or change to the current playlist, Some songs are just impulse adds.
Example: in 2000 a record promoter played a test pressing for me and my PD in San Diego. We asked to be given the test CD. Nobody had played it yet, and it was not yet released. The PD and I listened several times to the song, "A Puro Dolor" by an unknown Puerto Rican group, "Son by Four". We liked it so much we took it to the studio and said, "This is next". And we called the promoter who was driving back to LA and said "Listen right now. We are playing your song".
That song became the most played single on Latin radio for about the next 20 years. And it was an impulse add. I hope this anecdotal experience shows how things work "inside" the station.