> You can actually send from any third party address, e.g. Yahoo etc. It's literally what their page says.
Any third party _service_, not necessarily any domain.
> You don't have to own the domain.
Nowhere on the page does Google claim that you don't have to own the domain.
If the domain is already using Google for email, this feature will 'just' work. If the domain does _not_ use Google for email already, then it probably won't work, but that depends on the SPF and DMARC settings of that domain. For it to work reliably, Google must be added to the SPF policy, and the DKIM record must be created. If you do not own that domain, then you can't do that.
> I know their previous set up wasn't super safe, but they could have mitigated for this instead of just breaking the feature.
No, they couldn't "mitigate" against this. As explained, this is how email works, there is nothing that Google can change about this. As more and more domains are adopting DMARC (which is good practice), the more this alias feature will break.
This is why I called it a documentation oversight by Google. They should be clear about that the domain must either already be using Gmail, or that you need administrative access to the domain to add Google to the SPF policy and add the DKIM record to the DNS zone.