Agreed.
I know you asked the question of eeyore and I hope you won't mind me saying this.
What you suggest will work with the herbicide. If you can let the tree stand without any liability issues, it will provide much useful habitat as it rots. As the branches fall naturally, damage limitation is much reduced for the young standing trees. In about five years there will only be the stumps of the larger branches left.
If and when the trunk needs to be felled accurate directional felling will be less damaging than felling a newly dead or live tree with a "hard" crown.