Stock OS vs Custom, Proprietary Camera

The short answer is that like custom ROMs, it depends on which phone model.

The camera driver in the ROM implements Android's standard Camera2 API, which is common to all phones. Any app can use this API to access the camera. As you've pointed out, on some phones this requires having a binary-only camera driver in the ROM: without the binary part, only some modes may be available (e.g. lower resolutions), or the camera might not function at all. But if you have the binary driver, any app gets the full functionality of the Camera2 API, which is usually the full functionality of the camera.

The answer's not quite a flat "yes" because some phones have extra features. Phone manufacturers like to add extra camera features that aren't present in Android's API (yet). This mainly applies to the flagship phones from the big names like Samsung and LG, who compete on extra features not in stock Android. These extra features are only present in the manufacturer's own camera app, and they might use a special API to communicate with the camera driver, as well as Android's API.

These extra features will only be available from the manufacturer's camera app, and usually only on the stock ROM. Third-party camera apps will still be able to use the full functionality of the Camera2 interface: they just won't have access to these added features.

Note that image quality isn't just a property of the drivers. Many camera apps (both stock and third-party) do some amount of post-processing to make the image look better. You might find that a third-party camera app produces worse images just because its image processing isn't as good as in the stock camera app. Or you might find that a third-party camera app produces better images, if it uses an algorithm or technique that the stock app doesn't use.