If that is the primary reason they hired him, yes. His #1 job is the leader of the men in the locker room and to improve the product on the field. If he does that by improving the defense, great. If the defense doesn’t improve but the team does, that’s not an issue he should be martyred for if the team overall is showing clear improvement. If they only needed the defense fixed they could have just gotten a better D coordinator. If they needed the whole team on a new path, then they get Quinn to be the HC.
Let me redirect it back at you: In 1999 the Ravens hired Brian Billick, the architect of the historically dominant Minnesota Vikings offense. Then in 2000 he won the Super Bowl as HC, but they had a dismal offense to go with their alltime defense. Do you feel he failed them as HC that season because the team should have expected him to fix the offense first and foremost?
Of course not. The OC’s job is the offense, the DC’s job is the defense, the HC’s job is the team. No one feels Dan Quinn isn’t doing what Washington hired him to do by bringing the 18th best defense to the NFCCG. Everything is now his responsibility, not just the defense.