I think your assumption is correct. It does seem MapBox wants you to use their APIs and API keys.
However, there is a partial solution with raster tiles if you can fallback to that. Or you can get vector mbtiles from OSM data instead of from mapbox servers.
mapbox offline support for android/ios
I've contacted mapbox support in April 2017, and what I've been told is that although the style files that you can download from MapBox studio don't contain tiles, there's a way to have them render offline on Android/iOS.
If you would like to use Mapbox maps offline, you would need to follow the examples shown for Android or iOS, depending on your platform. Essentially, the mobile SDKs take in a style URL that points to your custom style created in Mapbox Studio, and the SDKs handle the downloading of all other resources needed to keep the map offline (fonts, icons, tilesets, etc). Check out our guide on offline maps to read more about how this works.
Links:
From a quick look at the example code, it would seem that the downloading of the tiles is done within the App, but done ahead of time, for instance just after you install the app and you still have
network connectivity. I don't immediately see whether or not your Mapbox API keys are involved at all in this download, with respect to your billing concern. There is a 6000 offline tile limit per offline region.
Not for desktop: The site also mentions that offline is not an option for MapboxGl-JS, so you'd have to hack it up to make your offline region available on desktop web browsers.
mapbox studio export as mbtiles...
There was supposedly a non obvious option in settings to download your source as an mbtiles file in mapbox studio classic. A mapbox classic issue on github has a screenshot of where the button should be. It's possible that option is gone now.
That github project, https://github.com/klokantech/vector-tiles-sample, provides links to projects/tools that can break the mbtiles file into its contained pbf files, which are a protobuf compressed format for each vectorial tile (https://github.com/mapbox/mbutil). Slightly outdated.
An approach which might otherwise work, is to cache the network requests ahead of time for each tile retrieved from MapBox, and then in your APP proxy the requests to your cache. I'm not sure whether that's against the ToS though. You'd probably have to do it programmatically with all XYZ coords you wish to cache.
Taking a dive with tippecanoe to produce vector tiles
You can create vector tiles with tippecanoe (and host them yourself). It's not MapBox, I know. But it might get your farther in the long run.
Nice short blog article about it: http://fuzzytolerance.info/blog/2017/02/02/Making-your-own-tiles-with-Tippecanoe/
The mbtiles you create, you can layer them, and possibly style them with the open source maputnik (which is somewhat like the style editor of mapbox). Then you take serve mbtiles with your own server, and plug in your style into mapboxGL-js or openlayers.
TileMill -- (.mbtiles with raster tiles) fallback
There is TileMill which is still available (but deprecated), runs on the desktop, and will allow you to create maps from shapefiles (e.g. you can get openstreetmap shapefiles for single cities at https://mapzen.com/data/metro-extracts/ ).
TileMill, will give you a mbtiles file at the end of the process. Although, that mbtiles contains raster tiles, not vector tiles. Vector tiles can be styled dynamically so it's nice on mobile when you want to accommodate various situations with fonts or adapt lighting.
Raster tiles might suffice, because in your app you probably will be able to render other interactive objects on top of that base layer. OpenLayers is one portable framework where you'd be able to do that, for instance -- allowing the user to draw objects or click points of interest that you've loaded from a database separately from the map data.
Get OSM-based vector tiles instead of mapbox
There seems to be an alternative maker of Vector tiles now, http://openmaptiles.org (used to be http://osm2vectortiles.org ).
You can go here, and download an mbtiles file for a city of your choice for instance: https://openmaptiles.org/downloads/#city , then serve that file (locally) for your app with a tile server. (or use one of the tools mentioned above to break it into the individual pbf files and read the files from local disk / memory)
With this approach, you combine a vector mbtiles from OSM, with the style you've created in mapbox studio (mapbox studio lets you download a zip file with your .json style in it), and you can still use MapBoxGL to display it.
There's Maputnik http://maputnik.com/ which you can use to edit a mapboxgl style file (the editor works with local storage). and there's presumably https://github.com/boundlessgeo/ol-mapbox-style which allows you to convert that style into a piece of code suitable for OpenLayer's style API.
