"overlooks our property" often means that the extension would be visible from the neighboring property, and that could be considered as being overbearing (which the planners seem not to have considered a serious issue). Overlooking from windows could occur, but the OP doesn't make this clear; in any case the planners decided that that wouldn't justify refusal.
I doubt if expensive planning consultants would make any difference in the case of a straghtforward domestic extension where the issues would be few and clearly assessable.
There is a presumption in favour of granting planning permission unless there would be significant demonstrable harm to interests of acknowledged importance. There would be no justification for refusal solely on the basis of a neighbour's objection, without significant demonstrable harm being caused. There has to be a failure to comply with local planning policy, not simply a local referendum; how would the OP feel if he proposed a perfectly reasonable extension and it was refused, the reason for refusal being that a neighbour didn't like the look of it, or a neighbour disliked the applicant personally.
It does seem that the issues would have been assessed and the decision issued lawfully. If that wasn't the case the decision could be challenged in terms of lawfulness. e.g. important issues were not considered, or proper procedures were not followed. Legal challenge would be costly, win or lose.