The current offerings of google, mapbox, bing, etc are prohibitively expensive when you just need a small subset of their features.
The company I work for is a happy customer of maptiler.com which even offers routing and reverse geocoding. It might not be as great and up to date as google's, but it's more than enough for our apps.
As I've said, the solutions are there, and there is an actual demand for cheap vector tile hosting without all of the other cruft that is bundled in the commercial solutions but isn't actually needed.
Fortunately there’s a large selection of background layers for leaflet: http://leaflet-extras.github.io/leaflet-providers/preview/ I quite like CartoDB Positron for as a muted background.
> However, OpenStreetMap’s own servers are run entirely on donated resources. They have strictly limited capacity. Heavy use of OSM tiles adversely affects people’s ability to edit the map, and is an abuse of the individual donations and sponsorship which provide hardware and bandwidth. As a result, we require that users of the tiles abide by this tile usage policy.
The default OSM tile server is not here to serve your startup.