Copyright covers four rights. The right to copy, the right to modify, the right to distribute, and the right to perform publicly.
The GPL places restrictions on a combination of the modification and distribution aspect, which says that if you modify the software and distribute it, then you must also include source code.
AGPL is more strict and says even if you just want to modify it and use it, you must supply the source code to the people who are using it. The important point here is that this is not triggering on use of the software (as EULA implies). It is only triggered after modification. And modification is something that is covered by copyright.
It's triggered on use of the modified software. Not just any use but use in such a way that network visitors interact with the software. If you don't use the software that way, the requirement doesn't apply even if you modified it.
If you've not modified the software, interested visitors can just get the source code from the same upstream, so why would it be required of you to host it.
However the following situation could arise and I don't see how the license addresses it. You visited a site running some AGPL service which was modified by those site operators. You decide to clone your own instance using their modified code. Since you're not modifying anything yourself you're not required to host that code. Now suppose that original site shuts down and disappears. The upstream for that modified code is no longer available. Are you on the hook for hosting it now? You've not modified anything yourself.
> 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. You are not responsible for enforcing compliance by third parties with this License.
Right, but suppose you disappear. The third parties are using code with your modification. Do they have to host anything? They did not modify anything themselves, but there is no upstream any more for the exact code they are using.
The GPL places restrictions on a combination of the modification and distribution aspect, which says that if you modify the software and distribute it, then you must also include source code.
AGPL is more strict and says even if you just want to modify it and use it, you must supply the source code to the people who are using it. The important point here is that this is not triggering on use of the software (as EULA implies). It is only triggered after modification. And modification is something that is covered by copyright.