The answer above pointed me in the right direction. Had to still do some research so I want to help the next person. Here is the solution:
Problem
After signing up for a Copilot Studio trial, the app hangs on an infinite loading screen. By default it redirects to a URL like:
https://copilotstudio.microsoft.com/environments/Default-{env-id}/home
This points at the tenant's Default environment, which often has no Dataverse instance attached. Copilot Studio requires Dataverse to initialize, so it never finishes loading.
Solution:
Create a new environment that has Dataverse enabled, then load Copilot Studio against that environment's ID instead of the Default one.
Check your Default environment
Go to Power Platform Admin Center → https://admin.powerplatform.microsoft.com → Environments → click your Default environment.
Look at the Dataverse column / Click + Add Dataverse on the Default environment page, fill out the form and save it. This should generate an environment apart from the default one. Now, get this new environment ID (it should not start with "Default") and replace it in the URL https://copilotstudio.microsoft.com/environments/{env-id}/home. Where env-id is the new one generated.
This is the solution that worked for me.