Just out of interest, is there a reason to not use the free wordpress plugin pretty links? This does what rebrandly does with the added benefits of using your domain out of the box, plus allowing you to see the stats inside wordpress (so you only have to be in one place).
No idea if such a plugin affects site performance though
There are some conspiracy theories that using your own domain as redirects can cause issues with Google. Rebrandly is widely known for its redirection and tracking capabilities so is less of an issue.
I don’t know if I buy that theory or not. I would assume if the redirect is using a 301 then its fine.