Given that mitochondria are evolutionarily closely related to bacteria, and antibiotics kill bacteria, why don't antibiotics also kill mitochondria?
The short answer is, as you found, that mitochondria are wrapped in a double membrane and are thus harder to penetrate with antibiotics.
The long answer is some of them do. Good antibiotics target structures widely found on and conserved in bacteria that are not found in human bodies. Some antibiotics are a little less specific and can indeed interact with other molecules. The allergy to penicillin, for example, is caused by the immune system reacting to a "novel" protein formed when penicillin binds to the surface of red blood cells. Some antibiotics can indeed harm mitochondria, but we tend not to use those and in fact wouldn't even call most of them antibiotics since we wouldn't use them because of their harmful effects. Here is a very recent report looking at some of the anti-mitochondrial effects of some antibiotics. To quote from the abstract:
We show that clinically relevant doses of bactericidal antibiotics—quinolones, aminoglycosides, and β-lactams—cause mitochondrial dysfunction and ROS [reactive oxygen species] overproduction in mammalian cells. We demonstrate that these bactericidal antibiotic–induced effects lead to oxidative damage to DNA, proteins, and membrane lipids.