That Apple employee is wrong.
I have a 9-year old iMac ( late 2009, 27 inch iMac ) and can run OS X 10.11 El Capitàn just fine.
I could even run macOS Sierra/High Sierra, but choose NOT to because my iMac has lots of older, legacy software.
If my 9 year old iMac can run OS X 10.11.6, your Mac can, too!
I installed iTunes 12.6.5 which requires OS X 10.11.4 or greater.
iTunes 12.6.5 works with iOS 12 and will still save both your apps and app data from the 6th gen iPod Touch.
You just need to upgrade your Mac to the latest version of El Capitàn or if this is not possible, to the last compatible Mac OS version, which is macOS High Sierra.
iTunes 12.6.5 found here.
https://support.apple.com/en-gb/HT208079
So, what’s the problem?
What is the actual Mac model you are using? Desktop or Laptop model?
2007/2008 iMacs can run OS X 10.11.6 El Capitàn as their maximum Mac OS.