Quite a lot of cache storage can be used and can be expanded on as well if there was a serious demand for it. Browser games don't technically have to redownload the entire game every load. The reason I claimed it as "cultural" is because of how we view browsers and interactive media within them. We are totally fine with opening a games store app, clicking download, and waiting and playing. The same is technically possible within the browser, but people are far less likely to wait for it.

It's like sound. People are OK with video games with clicky sounds going through menus with background music playing, but find it very obnoxious if a website does the same. Not a technical problem, but one of culture and how we view the mediums.

No it can't, for security reasons.

Browser games have no idea how long their data is around and how big the cache actually is.

You need to design a web game around the download bandwidth, and only stream data right when it's needed instead of preloading everything upfront. Porting existing games with a different streaming model usually doesn't make much sense. And once you build a web game from the ground up anyway, there's little reason to use one of the humongous game engines like Unity or Unreal which have been designed for a very different data streaming approach.