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 melt down in 3 minutes from direct contact with liquid. The amount of plastic you reduce is penny-on-the-100-bucks-note comparing to the amount of plastic waste produced by industrial activities.
The only way to fix the single-use container problem is for governments to ban it. Either the customers bring their own/rent the shops' containers for take away, or drink their beverages in the shop.
Is this doable? I guess. AFAIR the EU are experimenting with laws around this. Plastic bags ban is already visible in many country, even in non-first-world countries.