Fine, but personally I would like if the industry understood the principles behind these things. The OP might have wanted a yes or a no but he world rarely works thats way. Give a man a fish, feed him for a day, teach a man to fish, feed him forever.
OP asked if he could prune back to boundary. That's a quite extreme. You said absolutely not. That 's quite extreme too. Technically therefore me saying it isn't true is correct. The in-between scenario is that pruning can take place without TPO approval in specific circumstances.
The TPO relates to the tree, and it it's a group/area/woodland of trees it relates to all trees originating within that area. But it is a fundamentally important principle that statute should not stop anyone doing what he is obliged to do at common law, or under any other law for that matter. That is the basis of many of the exemptions. One can even see that when risk of damage is foreseeable but not imminent and a Council refuses permission for works to prevent damage, the obligation for compensation passes to it because it is preventing the tree owner from doing what he should and would, but for the TPO. That is only fair.
All this points to an important distinction. The TPO is over the trees, not the land, but restrains the owner of the land on which the tree originates. When the tree overhangs or sends roots into adjacent property, the tree owner is in 'tort' (or delict in Scotland) and he is not restrained from acting to remove the tort. Neither is the encroached party, he too must suffer the tree overhanging or growing under his property but only to the ponit where it is or is about to become a 'nuisance'. This usually involves damage but does not have to (see Williams and Waistell v Network Rail case a few years ago).
It is not, therefore a clear cut case of whether it is the tree or the property that the TPO affects. It is the tree up to the point where the owner of the tree is obliged or entitled by statute or common law to do something about the tree.