That would violate the license.

> However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so.

and

> Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License.

"Convey" is the key word there. You _must_ convey the license. Stripping it out, is not conveying it. Why is it automatic? Because under the AGPL, the license is a part of the work itself. You cannot remove it or modify it, without breaking the license, and thus having no right to modify it in the first place.

> Each time you convey a covered work

You didn't convey a covered work by using it to respond to a response or anywhere else in my hypothetical - and indeed that part of the license exactly matches the GPLv3. The relevant portion of the license is rather

> Notwithstanding any other provision of this License, if you modify the Program, your modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source of your version by providing access to the Corresponding Source from a network server at no charge, through some standard or customary means of facilitating copying of software. This Corresponding Source shall include the Corresponding Source for any work covered by version 3 of the GNU General Public License that is incorporated pursuant to the following paragraph.