I think the real issue is about whether your Apps Script project has been verified by Google.
In order to do that, you need to create a new GCP project and associate the Apps Script project with the GCP project.
The issue with using a free/consumer account is that you don't have an "organization" like you have with a Workspace account.
I think that part of the issue is that Google can verify and identify who you are.
If you sign up for a Workspace account, then you need to identify yourself, partly because you need to make monthly payments.
It's easy for a free/consumer account to be created without really providing information about where you live and who you are.
If you want your Web App to be available to the general public, then you'd probably want a Workspace account anyway.