All this greenwashing effort by big brands are laughable. The cost of actually recycle any used cups would far exceed the cost to manufacture them. When just the act of using those cups has already give them all the "environmentally aware" credit, why would they bother follow through with the rest of the process. The biggest scam is the paper straw. You still need a certain plastic liner, otherwise the straw will mel…
Bringing your own is a health hazard. It spreads germs all over the equipment.
I'm assuming you are much more conscious about this issue than I am (that's fine btw, people have various levels of germophobia) , but practically my whole office block do this without any noticeable health epidemic.
Also standard coffee shops (the Starbucks kinds) always have industrial scale utensil washers that rinse boiling water/UV radiate the utensils. Assuming a government issue the appropriate law, the coffee shops will be able to ensure your container hygiene just fine.