Leaflet JS is a generic library that works with mapbox and other tileserver apis. So yes you actually could do this trivially right now.
The API to tileservers is trivial, you provide a URL template and request the tile with the x/y/zoom you want and the tileserver responds with either a cached tile or it will render it on demand.
Good point. Leaflet could be another implementation option since Mapbox free tier gives you 50,000 loads of their JS client (unlimited tiles per load) and 750,000 raster tiles. [1]
So you could alternate loading Mapbox JS and loading Leaflet with Mapbox tiles.
The API to tileservers is trivial, you provide a URL template and request the tile with the x/y/zoom you want and the tileserver responds with either a cached tile or it will render it on demand.