I totally agree that the ideal folder would be the user's home folder.
I'm not at my SteamOS machine right now, so I can't check.... but many file managers allow you to enter the directory address manually. It might be worth checking into that until there's a solution. Then you could just enter ~/.
Which file manager are you using?
I am using Thunar. Your idea with manual directory navigation is great. I use it quite often, but I doubt steam will ever have tab completion, which is what saves me a lot of time defining the path.
Steam does not let you do that now, there is just the drop-down menu:
Unlike other file managers such as Thunar:
I wasn't sure if it tried to integrate with built in features of your actual file manager.
Nothing extracted yet.
When I open the Steam file browser (when changing my profile peciture for instance) it throws me to '~/.steam/steam/' by default. Clicking the 'up' button does nothing. I have to select '/' in the drop-down menu since there is no other option and navigate from the root directory to my home folder manually.
The browser crashed Steam for me a few times in the past as well, but I have no idea how to reproduce that issue. Maybe others are having a smilar problem.
I suppose the home folder of the user who started Steam could be used as the default directory by the browser.