It can be hard for greys/new players to get decent games. Getting kicked for being grey is a major negative in the new player experience, and it's not always easy to find a desirable 'all welcome' game.
Making global rating affected by TMM games would help to overcome this hurdle. It would also be sensible.
The calculations that determine the change in global rating from a TMM game could be based on the calculations using the players' global ratings (as though it was hosted in a rated custom lobby). Additionally, the relevant TMM rating would still be affected independently, based on its proper calculations (such as it is now).
Furthermore, if many people want to keep a separate rating system specific to rated custom games (what global rating is used for now), then I'd propose that global rating be duplicated and have the base version be renamed to something like 'custom rating'. Then, we'd have a 'custom rating' that functions exactly how 'global rating' functions now, and we'd have an actual 'global rating' that is affected by rated custom games as well as TMM games. Alternatively, we could stick with calling the current thing 'global rating' and make the new thing that includes TMM and rated custom games be called something like 'universal rating'. Regardless of what we call it, I think it'd probably be better to use the rating that also includes TMM games in lobbies to help solve the 'kicked greys' issue, among other things.