Are built in guitar tuners good?
The simple answer is built in tuners are bulky and aren't as accurate as external tuners because if it's built it it's going to be picking up both string vibration and the internal resonance of the guitar.
Do guitar tuners go bad?
The online tuner will give a pitch, and you can test your tuner with it. If dropped, it is possible for the oscillator crystal that it uses as a reference to become damaged. The typical failure mode is it stops ever recognizing notes, even though it turns on, if the crystal is damaged.