Symmetric keys means that anyone with the key can do everything the key allows one to do. That's bad if you're building consumer devices that contain the key. Signing with public key cryptography, which allows only verification with the end-user key, is much more secure in this scenario.
This isn't a good answer for "Why don't they use asymmetric keys for everything?" as you basically just explained that it would be more secure to use asymmetric keys.
