Both brake lights still out

Hello,

Since the third brake light is functional it means that the brake light switch is working.
So, the issue is likely a bad wiring.
The third brake light is on a different harness, so it works independently than the two main brake lights.
Good chance of a bad ground as both of them uses the same ground.

Here is a helpful guide of how to check wiring:

https://www.2carpros.com/articles/how-to-check-wiring

I have attached the wiring schematics below for your reference.

Please let me know of any questions.
Thank you.