I have a... heavily modified Browser 2 install on an insanely large (for a PS2) 1TB HDD. Between that and some other tools you can install and run full games onto the console. I even figured out how to make custom Browser icons for them, although in practice selecting one game out of even a decently-sized PS2 collection is ungodly slow. It's still cool as a retro hobby project, if you're willing to jump through the hoops to get a working setup.

AFAIK the official purpose of Browser 2 was to support PSBBN and distributing software through it. They never launched that service outside of Japan. Hell, most of the games that supported the HDD got that functionality stripped out of it in the US. PS2 games could install themselves to the hard drive for faster load times, much like the later install features on Xbox 360 and PS3. However, as far as I am aware this was exclusively for online service games in the US and Europe. On the original Xbox, Microsoft was already trying DLC and digital distribution (no, seriously, it had it, but you needed a special Arcade disc to run games). So I have to wonder how Sony wound up sleeping on something they were already doing for a whole generation.

There were a couple of games released in Japan that got the HDD support stripped out in the North American release. Two such examples are Star Ocean 3 and Xenosaga.

The PS2 OS is truly a work of art, such beautiful sound/graphics and ui design.