Why does Signal want a phone number to register if it's supposedly privacy first?
I remember a time when visiting a website that opens a javacript dialog box asking for your name so the message "hi <name entered>" could be displayed was baulked at.
Why does signal want a phone number to register? Is there a better alternative?
No it isn't a URL. But that would indeed be the way they could make it work. If they did that, then...
If you don't have the app installed it installs it from the web site. If you have it installed then the app takes over instead of the web browser. That is how many apps work (eg Reddit).
You can configure one or more of your profiles' addresses to be a "business address" which means that when people contact you via it it will always create a new group automatically. Then you can (optionally, on a per-contact basis) add your other devices' profiles to it (as can your contact with their other devices, after you make them an admin of the group).
It's not the most obvious/intuitive system but it works well and imo this paradigm is actually better than most systems' multi-device support in that you can see which device someone is sending from and you can choose to give different contacts access to a different subset of your devices than others.