Between the two, which is a must have? Which is a good to have?

My answer is, there is no good balance. You sought both, you get nothing. One has to make some way for the other.
However, if you are very highly effective, there will be rub-off engagement, vice versa is true.
In my previous school, engagement is the key to effectiveness, now, effectiveness is the key to engagement. 
A fellow colleague from another workplace, told me his boss value , engagement a lot, care a lot of his staff welfare. Nope, it is not lip service, all meetings must end by 3, all official events must end by 5. While I spend 2 weeks after the school holiday starts for review and planning next year work plan, their staff are off to their holidays just 3 days in the holiday. My colleague admits her staff are spoilt …