The correct tools are those small plastic containers or a cheap grease gun. You can get the grease better into the bearing with them and don't have the risk of smearing cotton fiber in there. They are of course more expensive than a q-tip, but you can get one for unter 5€. Seriously, if you do this even a couple times a year, buy one.

(This is also a great example why environmental regulation is so tricky: It totally makes sense to prevent one of the worst polluting product to be phased out or replaced with a better solution. But then there are edge cases (how many people have even greased a bearing in their life?) where the new product might be worse, but that still is not an argument for mass pollution on our beaches or against that regulation)

I can find both paper stick and plastic stick next to each other on the shelf at my local grocery.

It must be the off brands then. I dunno, I don’t buy that junk but I have noticed it in other people’s bathrooms.