One possible solution would be to use the Thunderbird email client and configure it to use a single shared account ("Local Folders") for all of your mailboxes. All of your new mail would show up in the same Inbox folder, a copy of all of your sent mail would be in the same Sent folder etc. You could still send/receive mail for all of the mailbox's email addresses but you would only have to configure settings for one account (the shared account), making it easier to manage.
There is no limit to how many email addresses/mailboxes you can do this with in Thunderbird. You can configure the folder listing to add a column that identifies what mailbox the message was sent to.
If you wanted to do this you would add a POP account for each mailbox and then reconfigure each to use a "global inbox". That would hide each account folders and have it share the folders in "Local Folders" (a special account meant for sharing mail from multiple accounts). See
http://kb.mozillazine.org/Global_Inbox for details.
Alternatively if you prefer webmail you might be able to configure it to automatically fetch new mail from other addresses and to use any of those email addresses when sending. The problem is that many restrict how many aliases/mailboxes you can add. Outlook.com supposedly only supports 10 aliases for example.
https://www.lifewire.com/create-alia...utlook-1170647