Not sure what directories you're referring to as "listed above", but in all my HostGator accounts, the document root for the web server is public_html. So everything you want to serve to the public via the web needs to be somewhere in that directory.
None of my accounts have any "home" directory - not sure what that is, but my guess is, it might be a symbolic link to the public_html directory. If that is what it is, then you will see everything that's in the public_html directory in home, as well.
You can either install C5 directly into public_html, or create sub-directory to install it in, if you want to keep C5 isolated from any other systems you may want to install to that site. I don't usually use the automatic installer, as I also have had issues with it in the past, and it's not that much more work to install C5 the normal way. That is what I would suggest.
The home directory is just the one that shows in the cPanel file manager; All the Hostgator sub-directories fall under that including /home_sitename/public_html. I attached a file showing the structure from cPanel (/home_sitename is hidden at the top in that view).
In my install, public_html/Concrete5 is where some directories are found. There is a further public_html/Concrete5/concrete sub-directory that has most of the directories listed as directories that are supposed to be in the root directory as seen in the Concrete5 help document at the top of this help page.
So I assume, in this case the /Concrete5 is the root directory and that is where those directories belong.
I've had to add directories to /Concrete5 directory to get various things to work - themes, jobs, etc.
I think I'll try your install method for 5.2.2 vs the Hostgator install for the upgrade and see if that works better.
I see... the way you had them shown in your original post, the directories all appeared to be at the same level. Kinda changes the meaning of what you're saying. But, it sounds like you have it figured out, so all is well.