Build system: validate that documentation doesn't have any errors or warnings during PR validation

[Aaronontheweb]

To help ensure clean DocFx hygiene, we need to ensure that we're not leaving behind broken links or other issues such as #5527, which was introduced when we changed #5500. Therefore, I think the build system needs to flag this by building the website upon every single pull request since merges into dev usually trigger website updates.

We can manually authorize this stage to determine if it's needed for a particular PR, but we need some automation to check against this.

[Aaronontheweb]

Another PR we wouldn't have had to made if we had this: #5532 (review)