Actually, I think it does not. I think you would be in compliance with AGPL if you had an internal server run the AGPL program and then had your external servers communicate with the internal AGPL running servers to perform work. As long as the web user was not communicating directly with the AGPL, the user to whom you would have to release your modified source is yourself since the outside user never communicates directly.

Next, the AGPL is and will be a very small percentage of GPL licensed software.

Finally, apart from making a political statement, I do not see the AGPL getting any traction. Apart from MongoDB (which has its drivers as Apache) I cannot think of any widely used software that is AGPL.

Also, note that while MongoDB is AGPL, if you look at the contributor license (https://www.mongodb.com/legal/contributor-agreement) any contributor is giving MongoDB the right to use and license your contribution and to make derivative works of which they are the sole owner. They can also license your code under a non-copyleft license to others. I do not see how in this case a (A)GPL is more free than a BSD/MIT license. Along this line, you will see that many successfull GPL projects (excepting Linux and the FSF ones) are of this type where the source code is GPL but any contributions give some type of copyright assignment to the company which then sells basically closed-source licenses to corporations who don't want the GPL.

edit: Rewrote first point.