At least in Australia the lids are considered too small to recycle as they are too light and blow around so you’re supposed to put them in the rubbish. They had also traditionally been a different plastic which sometimes was an issue.

So it does truly potentially help in the recycling regard. Note however the professing facilities and requirements for this vary wildly from state to state and country to country and also they often ask for a lowest common denominator of all the recycling plants. So it’s not that some places potentially can’t but not all.

Leaving them on also causes issues because if the bottles are air right they don’t compress as well.