Having same issue with AT&T. Got the Watch 7 a couple days ago and spent a lot of time with AT&T via chat. At the end the guys said to wait for 3 hours and then it'll work. Replaced an existing line with a "watch line" on my plan, which would save $5 / month. Hasn't worked yet, and I have all the results OP mentioned above. Watch connects with my iPhone 11 fine, but no independent cell connection for watch. Also, a new line was added to my account (which this person saw as an iPhone 13 Pro (never had one), and the AT&T App shows this new line is "Device Suspended". that's the line to which nothing has ever been connected or activated.
Am now 66 minutes into a call with AT&T, and they don't have a clue. Like, this person has no idea what they are doing. Has put me on hold 20 times. At one point they said they would send me a SIM card, but I said "Wait, I don't think these watches have SIM cards. I think they use eSim cards." That triggered another 15 minutes. Final answer: I'm supposed to go to a physical AT&T store to get an eSIM for my watch to work.
This is after confirming and reconfirming Sunday that my guy on the AT&T chat hooked me up. Just had to wait 3 hours for it to show in the system. Today another hour on the phone speaking with someone, and the answer was go to a physical store to get an electronic SIM? Pure craziness. Everyone else I know got their watch, paired it with their phone, and there was no hassle with the cellular service on the watch registering. Not my first time dealing with AT&T hoops. There's no good reason for this, it's just a matter of calling again and finding someone who knows how to make this happen.