Phoenix basically did what the Celtics chose not to. Kevin Young was the favorite for awhile (to what extent is debatable), until they decided to go with the more experienced Vogel. To keep Young on staff, they made him the highest-paid assistant in the league. You have to wonder if they looked the the Celtics' season and realized a first-time head coach was not the right choice for a team that has to win now before Durant is too old. Durant is going to be 35 next year, after all.
The Cs could have hired Vogel last year, he wasn't working. Mazzula could have been promised the lead assistant job when Ime's guys left at the end of the season. They didn't do it, for whatever reason. Maybe Vogel thought it was too sudden and/or wanted a break, though you'd think it'd be really hard to turn down the Celtics (Vogel also started his career with the Cs during Pitino's regime). Since they clearly wanted Vogel as an assistant this season, you have to assume they like him and were comfortable with him as a head coach too.
Other factors could have been:
1) Money - As long as Wyc is in charge, money is always a factor. He may not have wanted to pay three former head coaches at a time (Brad, half of the suspended Udoka's $4m salary, and the new coach's salary). It seems insane to worry about 2-5 million dollars if you're paying 70mil in luxury taxes, but who knows? I don't know if Vogel was still getting paid by the Lakers after getting fired, but it's possible he would have forfeited that if he took on another job. It would have to have been worth his while if the Celtics weren't offering something commensurate (this is why Detroit had to pay Monty Williams so much - he was still being paid by the Suns and would have collected paychecks for 3 years as long as he was unemployed).
2) The Celtics made a big deal a year ago about hiring a black head coach. The situation was already a PR nightmare, but it would have looked even worse if they suspended/fired him after one year and replaced him with a white coach.
3) Vogel is a defensive guy, like Udoka. The C's problems last season were on offensive turnovers/execution/playmaking. Did they think Mazzulla as HC would improve the offense?