The issue isn’t really the paper (although one can argue that any single use item isn’t ideal), paper coffee cups contain a thin layer of plastic on the inside to prevent leaking, which prevents them from being easily recycled or biodegrading naturally (which also isn’t possible in a landfill). A minority of recycling plants actually has the capability to separate the paper from the plastic, but this is not time or cost efficient and so it’s easier to just throw it out.

Also, some places do brew or mix in a paper cup first, some will brew directly into your cup.

In the UK we have the capability[1][2] to recycle all the traditional PE lined coffee cups we use here.

There are also quite a few collection services[3][4]. One of the challenges is separating the waste ready for recycling so having collection points in cafes, offices, and railway stations[5] is a really good start.

There are also some new cups with reduce the amount of plastic in a cup so that it's below the 5% contamination level fr general paper recycling. Personally I think this is not a solution as it's just gaming the system buy encouraging contamination of the waste stream.

[1] https://www.dssmith.com/recycling/insights/case-studies/case...

[2] https://www.recyclingtoday.com/article/james-cropper-paper-m...

[3] https://thefirstmile.co.uk/service/coffee-cup-recycling

[4] https://www.veolia.co.uk/services/coffee-cup-recycling

[5] https://www.networkrail.co.uk/news/network-rail-introduces-c...