I am running Apple Mail on my Mac
(not the distinction) and this is not a problem for me.
I have 5 mail accounts, all of which are displayed in the Mailboxes section in the left hand window.
By clicking the expose triangle, I can either see all the accounts separately or all lumped into the one Inbox.
When I see them all separately, if I select one, all I see in the Inbox is mail from sent to that account. Ditto for the Sent, Trash and Junk folders.
No sure why all your accounts should be in the "On My Computer section, unless it has something to do with MAPI and POP accounts. (a suggestion made earlier)