Gmail – My primary Gmail has several other Gmail accounts connected. One of them is sending the primary a copy of every outgoing message it sends

I use the following setup:

Under Settings -> Forwarding and POP/IMAP, you can forward email from one account to another.

Under Accounts and Import, you can add additional email addresses using "send mail from another address." Select "Reply from the same address the message was sent to" to have Gmail pick the correct address automatically when you reply.

With this configuration, you will receive emails addressed to both accounts and can send from both accounts.

I think there are a few things going on here.

By setting up secondary emails so that you can send "from" them, I don't believe you will be able to really break the link that others are seeing between those email addresses and the primary one that is associated with your Google+ account. You are essentially merging aspects of these various email addresses so in a sense they are becoming one identity.

Maybe a better way for you to gather all of your incoming mail in one place would be to setup forwarding from the secondary accounts rather than pulling of the secondary accounts into the primary account. This would not allow you to send "from" those accounts though. If you really need to keep more of a separation of identities (for various aspects of your personal and/or business life), I think you may want to setup an additional primary Gmail account which can be used for the additional identity. Then, for example, any business email addresses can be associated to that secondary identity and any personal email addresses can be associated to the identity you currently have (the one with the Google+ profile that you don't want to advertise so widely).

But something else to consider is that even if you completely break apart your identities, as long as you have a public Google+ profile people will be able to find whatever you have chosen to make public. Maybe you also want to consider making private certain aspects of your Google+ profile?

Specifically, there are settings you can apply for your profile picture's visibility. Please see the Stack Exchange question How does "your Gmail picture" work? for some more details.

Hope some of this helps...