Even if you mounted the filesystems, you'd still have to retrieve all the 1s and 0s over then internet, so your speed wouldn't be any higher (in fact I'd consider it likely that standard kernel filesystem stuff isn't as optimised for HTTP over internet as actual internet utilities like browsers).
If your browser's *really* bad at downloading, you could use something like
wget to retrieve the remote file - no frills, very low system load, and it just works.
Or am I missing something here?