You shouldn't be able to play a game while it's downloading, it's an EXE that needs to be executed in order to be installed. I don't think an existing game would work either because it's still an EXE that patches the install so it should have to complete before it can even execute. It's not the same as music and movies that can be streamed.
some consoles and PC game clients (Origin does it) prioritize the download to get the game executable installed first, then download the large content files in such a way that the ones that are needed early on are installed first, allowing the game to be started while the rest of the files are still being downloaded in the background. Obviously this won't work well in all scenarios, and requires the devs and publishers to set up the content packs with this feature in mind.
Anyway, no, Quest doesn't do this. I don't think it will even allow any downloading to happen while a VR app is running.