Germany has a recycling rate for PET bottles of 94.8%, 45% for general plastic packaging. It's profitable because if you sell stuff in plastic packaging you're forced to pay for the recycling. As a consumer you don't pay for having those yellow bags and bins collected, which means that if you separate all the packaging and put it in there you get to pay less for your other bin. As a producer, if you're switching your packaging to something that's easier to recycle, you pay less into the scheme and either lower prices to sell more product or pocket the difference. Our Greens love to solve shit by strategically shifting around market incentives: When the producer has to pay the bill the market failure suddenly vanishes.

Glass OTOH is a problem because even small contaminations can mess up a whole batch, and the process is energy-intensive in the first place. Crushed glass makes very good aggregate for concrete, though.

As to cardboard: Sort what you can, the rest goes into industrial composting. Left-over plastic gets burned, which actually isn't that bad -- the issue isn't the burning (they've got proper filters and everything) but that the plastic isn't sourced from renewable sources. Certainly more sensible even in its current state than burning oil while putting the plastic in a landfill.

Mind sharing a source and saying what country youre talking about? Recycling is most certainly not a lie. Yes, some countries dont give a fuck and send it all to a landfill, but others dont.
There is also the fact that there are more and more battery processing plants too that separate the materials for reuse. Plastics can be remelted and reused after they are grinded and separated. And cardboard, afaik, can just be reused. The ink on the cardboard is just merged into it. There is a reason recycled cardboard has a distinct colour and texture