Hi, founder of Geocode Earth (https://geocode.earth), another geocoding service that _does_ use OpenAddresses and only other open data sets.
Mapbox purchases data from, I believe, TomTom, and other proprietary sources. That's why they have cheap(er) pricing if you want to get results without the legal ability to store them, and much more expensive pricing if you do want to store it. I'm sure negotiating that arrangement with the proprietary address data vendors was...fun.
A friend of mine runs opencage, which is a geocoder based on open streetmaps and a few other data sets. They got a lot more business when Google raised their prices a while back. Anyway, go give them some love if you need a geocoder because they are an awesome small company with a great service. Their freemium layer is also pretty generous.
Nominatim is the defactor open streetmaps geocoder but its weakness is that it relies on only that; so it is only as good as OSM is.
Geocoding is a surprisingly hard problem to solve with and without open data. Picking apart text strings into unambiguous addresses is just super hard with all the conventions for this around the world. One area open streetmaps is still inconsistent is house numbers and postal codes. Getting solid open data for that is just hard in some places.
Mapbox purchases data from, I believe, TomTom, and other proprietary sources. That's why they have cheap(er) pricing if you want to get results without the legal ability to store them, and much more expensive pricing if you do want to store it. I'm sure negotiating that arrangement with the proprietary address data vendors was...fun.