Why won't my iPhone 13 connect to iTunes on my iMac?

I love iTunes, always have. And I still have a HUGE library of music in my iTunes on my old iMac. But I think my iMac is now out of date. It still works but several websites no longer work at all on it. I'm now getting messages that my iMac OS needs to be updated. I also recently bought a new iPhone 13.

When I plug my iPhone 13 into my iMac... the iPhone does not open on the screen as it should in my iTunes program. My last phone, an iPhone 8+ always did open up in iTunes and it would easily update and add new iTunes playlists. But now my iPhone 13 will not open in iTunes.

I'm wondering, is the issue that fact that my OS in my iMac is now officially out of date. My iMac is running OS X 10.11 and the messages tell me I need MAC OS 10.13 or newer

[Re-Titled by Moderator]