Mayo/Aioli work great for really any dish that you’d use egg and oil in anyways. I tend to use it when I’m making condiments I want infused. Bread too. Meatloaf works great with mayo.
I’ll use mustard in a lot of things, but not actually for emulsion. I just really like mustard lol. Garlic and onion too actually. They’ll help a sauce to come together slightly as well. In some cases I’ll be failing to get a sauce to keep from breaking and then I add some garlic paste to the sauce and stir it in, heat it gently, and then let it thicken. It’s usually done after that and I didn’t use any flour to make my gravy or whatever else keeping more flavor intact.
Edit: yes I did say I’ve found garlic paste to fix a broken gravy/pan sauce. Without flour.