AFAIK, the main issue of the GPL family of licenses is that they are rather poorly drafted. The GNU project has great evangelists but ideological-minded people make bad lawyers. Their licenses are long winded and hard to interpret, and often make assumptions about unsettled bits of copyright law. Their legal theories (eg. "GPL is not a contract") have been rejected by courts.
In short, if you're a company wanting to use some AGPL-like license, you'd probably want to use a license that's better drafted over the original AGPL.
If it gets to court you've got a plaintiff either claiming that defendant violated copyright or the defendant failed to follow the terms of GPL.
In both cases the defendant will respond that they followed the terms of GPL.
In both cases the court has to determine what the license means and then whether or not defendant followed the terms of the GPL. To do that the court has to decide what those terms actually mean.
For that I don't think it matters if GPL is considered to be a "contract" or "license" because as far as I have been able to tell the rules of interpretation are the same no matter how you characterize the document.
Maybe it could make a difference in the remedies that are available if plaintiff wins. If it is seen as not being a contract then it might be that the only remedies would be those under copyright law. Those would generally be an injunction ordering defendant to stop infringing and monetary damages. Actual monetary damages would be near zero so plaintiff would probably ask for statutory damages.
If it is seen as being a contract then potentially the remedy could include an order that the defendant release their source code. If the GPL infringement was distributing binaries without making source available this would probably be the remedy plaintiff wants.
Ordering the defendant to obey the terms of the contract is called "specific performance" and I believe is usually disfavored by courts if monetary damages are sufficient but since infringing GPL generally doesn't cause any monetary losses to the copyright owner and the whole point of the contract was to make source available I'd expect a decent chance plaintiff could get specific performance.
Note that this means that it is actually better for GPL to be seen as a contract.
That the GPL is a contract hardly came as any surprise to people familiar with the law. Lawyers and judges view any collection of terms as a contract, and tear-open licenses are the norm these days. But the FSF had its own reasons to say it’s a license, reasons that might be more important to the philosophy of Free Software than the court.
Contracts require consent between two parties who join in the contract, if they are to be enforced. The GPL (and many other licenses that come with products) doesn’t have a signature page, so there is no explicit consent. If there is consent at all, it’s the implied consent that has become standard for “tear-open” licenses. Your acceptance of the license is indicated by some action, in this case integrating the code into your product.
The default under copyright law is all rights reserved, which means “you can’t do anything with this” with some minor exceptions that are called “fair use”. Thus, if you integrate the software into your product, distribute it, or perform some other action that is restricted by copyright, you must have accepted the license because your alternative would be all rights reserved. Thus, the FSF asserts that the GPL is a license because they feel consent isn’t really necessary, they don’t want to argue about consent in court, and they believe that they can do all of the enforcement they need using a complaint of copyright infringement. Also, tear-open licenses were a much more foggy issue in law when the GPL came about. And then there’s the philosophical matter:
Historically collecting damages were not the goal of GPL enforcement. Specific performance, or stop infringement by no longer distributing the software were the remedies. The latter is not always an option for the infringing party, and that's what submits them into compliance.
In short, if you're a company wanting to use some AGPL-like license, you'd probably want to use a license that's better drafted over the original AGPL.