All I see at https://service.tesla.com/service-manuals is the manual for the legacy Roadster, which Tesla no longer supports.
And if I login, all I can see is the option to buy some kind of service subscription.
You need to click "Model 3" (or whichever other model you were trying to find a service manual for) in the top nav bar. If everything went right, you will see this screen[0], with all the needed links (instead of what you saw previously).
Once signed in, I was able to hit “Menu”, then select Model 3, which took me to that page.
On that page, I can see the service manual and more, for free.
Do you know of companies charging for service manuals?
Tesla and these guys are the exceptions.