Flash sockets don't transport HTTP requests. You might be able to tunnel them through WebSockets, but that'd turn the "helper server" into what would amount to an open network proxy -- it'd be trivial to abuse.

It's kind of a moot point, though; I doubt that the networking servers that these games depended on are still online. That's a whole separate preservation problem.

No need to make it an open server. For applications that require a limited number of resources, you can implement a whitelist--only allow connections to certain IPs or addresses. Inspect the connections if you have to. For everything else? Make the user run it locally, have your runtime try to connect to a local instance before connecting to a remote instance.

Alternatively, figure out a way to ship remote dependencies alongside the swf file, and define a patch format that allows replacing network requests.

All of this is more work to do, but all of it is also workable solutions.

The worst part of that preservation problem is, that as a DRM measure, some games will "phone home" to download a swf component that implements some critical path of the game. If the server goes down, those games are dead in the water unless you can reimplement the missing pieces.