Would it be possible to have browse() be much faster simply by using browse_rsc() when the person logs in to cache every single file for them?
Wouldn't make any difference, browse_rsc() isn't what causes the file to be cached, logging in and downloading the game's resources does.

By that point all browse_rsc() is doing is setting up the cached file so it can be easily accessed by the client's browser.
In response to Nadrew (#1)
What I'm trying to say is that I will have hosted files online, and would like to cache them before they are actually needed. Wait a second...I see how flawed my logic is at an attempt to save space! Thanks anyways!