The tech support script is there for one reason -- to let the higher-tier support staff who have actual problem solving skills work on actual problems without wasting their time on people who need just need to be told that the "any" key isn't literally a key. It does this by letting people of lesser ability handle the easy stuff and -- this is key -- letting the upper tier know that the easy stuff has already been checked.
"But I've already checked the easy stuff. It's plugged in, I have tcpdump output, I can prove that it's an actual problem!" Maybe, maybe not. Think of it as an input validation problem. A web server should never implicitly trust what the browser sends, right? It's poor practice to let the browser do all the input validation and blindly accept it. The script is the tech support input validation step. You say you're an advanced user who's tried everything easy, but how does tech support know that? Just like 90% of drivers think they're above average, 90% of geeks think they're above making stupid mistakes. The very fact that you're quibbling over the terminology of "modem" vs. "router" makes be believe that you're someone with an over-inflated sense of their own abilities.
Let them run through the script. It's tedious. It's frustrating. But it does tend to check the stupid "is it plugged in stuff" that even the most tech-savvy can sometimes forget. When they ask you to do stuff, actually do it. Don't just say, "Okay, I'm rebooting now" while you sit and play Cookie Clicker for five minutes pretending to do it because you "know" that's not the problem. You might get surprised and find that your problem isn't as exotic and unique as you thought it was, or that your list of "everything" to try doesn't really include everything.(*)
And if you do get to the end of the script without fixing it, the upper tier support person will have reasonable confidence that you do indeed have a non-trivial problem.
(*) Personal anecdote: My wife's laptop had a flaky USB port. After checking all the easy stuff I got on the phone with tech support. "Turn off the computer, remove the battery, and hold the power button down for 60 seconds." What? That's ridiculous! There's no way that'll fix it! But I did it anyway. Guess what? It started working, and has worked flawlessly since then.