Based on the single stream comment, I'm going to assume you're describing an experience in North America, and your assumptions are pretty off base for PET bottles.
Figure 1 in the linked paper gives the raw numbers for where PET bottles end up. [1]
Yep, I've worked in recycling for 6 years and am well aware that there's a lot of broken parts. My point was that if a PET bottle goes into a single stream system, the recycling industry is pretty good at capturing it, baling it, and selling it to a plastics processor. That processor will clean it, pellet it, and sell it as rPET. 85% of PET bottles that end up in curbside single stream end up getting recycled. I'd like that number higher, but it's where things are at the moment. Throw a bottle in the trash, it's getting landfilled/burned. Throw it in recycling, it's most likely getting recycled.
In Australia the whole hard VS soft plastic blew up recently. The REDcycle brand by mayor supermarkets was shown to just collect and store the soft plastic in large warehouses.
For bottles there is a refund scheme.
Figure 1 in the linked paper gives the raw numbers for where PET bottles end up. [1]
[1] https://onlinelibrary.wiley.com/doi/10.1111/jiec.13496