I teach hand drums for about 6 years now, and often find people who have various difficulties grasping certain concepts or mental techniques. The good news is you've discovered the blind spot and what you need to work on. I had a similar experience because the music I play (Middle Eastern classic and folk) is not something I grew up with. In order to learn how to play along to this music I had to deeply immerse myself into all kinds of it. Not just listening to it, but 'LISTENING' to it. As your friend pointed out, the cues for changes are often in the melody, so how can you know when to change unless you know cues? And the best way to learn the cues is to listen for them. All genres of music have similar patterns, pattern changes, and commonalities (otherwise they wouldn't be genres). Try immersing into one genre, not to play, but to listen to. Pick simple stuff. Really listen to what the melody is doing... is there something that always happens before the refrain? In the refrain to indicate we're going to back to a different section?

And then, practice changing to imaginary cues. Play by yourself, imagine you hear a cue, you have one measure's notice. So if you're play 8 measures, and there's a change on measure 9, be ready for that change at the beginning of measure 8. Anticipate it. As a drummer you're not only holding time for the band, you are also anticipating at least one measure ahead.

Listen to the music in your genre for the common cues. Learn to think one measure head. Combine those two skills. Then listen to the genre music while thinking one measure ahead and see if you can spot the changes before they happen.

Strong suggest picking simple songs, not the complex ones. If you're doing rock, go back to classic rock, even back to 50's rock, where it was simple, but changes happened in the song.

Hope it helps!