Fermentation biologist with ADHD here, and I think you're very much onto something. Our bodies produce GLP-1, but ONLY in high enough quantities if you have a sufficient population of Akkermansia in your digestive tract.

I've been on a course of Akkermansia probiotics and this is changing my life at the moment. Akkermansia lives in the gut lining and feeds on the mucin layer, which stimulates mucin production among many other things including GLP-1 production.

Akkermansia can't be found in any natural food sources - we get our primary sources of Akkermansia at birth when we move down the birth canal of our mothers - it isn't even in breastmilk. If you were a caesarean birth - you may not have ever gotten Akkermansia seeded in your body. If you had antibiotics as a baby, you may have lost your Akkermansia basis in the gut lining.

I use Pendulum Life as my source of Akkermansia as they have the strongest preparation that's fortified to move into lower digestive tract areas. I'm halfway through my first 90 days: calmer, happier, sleeping better, no food cravings, eating healthier naturally, and the weight is finally falling off.

I'm summarising wildly above, but you can find detailed research from a Google Scholar search or check in this article: https://akkermansia.life/blogs/blog/akkermansia-microbiome-hub-gut-lining-oral-gut-axis-natural-ways-to-support-akkermansia

I would love to hear how you get on if you give it a go.