Here's the 2021 annual report from RIRRC who handles all waste and recycling for the state.

Bottom-line figures, statewide totals for the year:

Solid waste: 374,818 tons

Recycling (including cans and bottles, compost, scrap metal): 185,111 tons

Rejected recycling: 18,744 tons

Presumably, that means that 185,111-18,744=166,367 tons actually gets recycled, but I am not finding data specifically about how much recyclable waste actually gets recycled. We seem to have state laws requiring recyclable material to actually be recycled, but I am far from a legal expert and there are surely loopholes; § 23-18.8-2 (3) "All solid waste capable of being recycled should be recycled".

Here's a story from the ASRI, which includes a description and picture of the sorting facility you were curious/skeptical about: https://asri.org/news-events/articles-2020-01/the-truth-about-your-recycling-bin.html