This is madness right?

Yes, sounds like it

  My real main issue is that time just flies by and I am learning nothing and accomplishing nothing

If you are constantly bustling between meetings, needing to context switch between teams/products/features, it will make it more difficult to make headway/improvement in any one direction (sense of stagnation)

  The advice given to me was to make a mini plan for my own time and to define the role I am in myself. This confuses me because it doesn't actually solve my issue that my time is spread too thin.

Sounds like you can give yourself a title promotion, how about Principal Software Architect :) (just kidding). More seriously, one suggestion (other than reducing scope of responsibilities) might be to have the 3 teams nominate a go-between; you could then do a daily sync with each team's nominee and hopefully these syncs could reduce the amount of time spent you need to sit in the meetings yourself (to help free up some bandwidth, at least)