This presupposes that modification of a copy of a copyrighted work that one lawfully owns is, in itself (even without subsequent distribution), an act restricted by copyright. I contest this assertion.
Even if it were the case, the permissions granted by the AGPL are not mutually exclusive with any other set of permissions. Dual licencing, for example, is a common practice. Thus, I'd argue one can take advantage of provisions of the AGPL permitting modification, but not rely on it, but instead on the provisions of copyright law itself, to actually execute the software. The AGPL is not a contract, it is a copyright licence; as such, the set of actions you can take for an AGPL-licenced piece of software is by definition a strict superset of the set of actions you can take for "all rights reserved" software.
Under US copyright law the "preparation of a derivative work" (in this context more or less making a change to the code base) is one of the six exclusive rights under the control of the copyright holder [1].
Even if it were the case, the permissions granted by the AGPL are not mutually exclusive with any other set of permissions. Dual licencing, for example, is a common practice. Thus, I'd argue one can take advantage of provisions of the AGPL permitting modification, but not rely on it, but instead on the provisions of copyright law itself, to actually execute the software. The AGPL is not a contract, it is a copyright licence; as such, the set of actions you can take for an AGPL-licenced piece of software is by definition a strict superset of the set of actions you can take for "all rights reserved" software.