While you are simply able to Check out then check back in without any problems on the web access (without downloading or changing the file), you are not able to check out, download a copy make changes then check back in the edited copy.
this is why we are using document connection to manage this process on a mac, the temporary file resides in the library document connection cache for easy management for multiple files from multiple and various locations on the team site.
in your test as you details above, did you do this on a mac? did you check out, download the file, alter the file, save the changes then try to check it back in with changes on the file?
The potential solution or workaround you are trying to state above possibly would be, check out, download copy, edit copy, upload and overwrite with check in. it is not an enterprise solution, it may work for 1 document here and there, but not as a working solution.