The major game engines include networking.
* Unreal Engine, networking section, documentation.[1]
* Unity game engine, networking section, documentation.[2]
My own interest is in metaverse systems, with big, detailed, free-play worlds. Wrangling all that content presents a set of problems that are not well solved yet. Which is why the highly publicized metaverses look so much worse than a typical AAA title. They haven't solved content wrangling, so they have to limit content complexity so severely it looks like 1995. I'm plugging away on that problem. (Demo video: [3]). It seems to be quite solveable, but there's not much written on the subject and I'm having to re-invent too much.
[1] https://docs.unrealengine.com/5.0/en-US/networking-and-multi...
If your team decides to create singleplayer game(s), and your own engine as a platform for it, then building full networking and netcode into that would be out of scope. That would still be a game engine nonetheless.
Can the term game engine include networking? Absolutely. Does game engine inherently imply networking? No, I don't think so.
If "controls" are an integral part of a game engine, then networking is as well. Networking is just another control/input device. You read bytes from your joystick driver, you read bytes from your network driver.
Building a robust input handling system that can handle a variety of input devices, high latencies, failures, recoveries, etc is absolutely an integral part of building a modern game engine.