One can prune lilacs. I know that there are some leaves on mine which look suspicious. I aim to cut them off today or tomorrow. Also I will cut off the remains of the flowers. They say it is best to prune just after flowering, but we know there are some late flowering lilacs, like the Ivory Silk. They also say not to prune too heavily, but if you are considering removing the tree, surely a good pruning would be a better alternative. You know the situation better though.

It would be nice to see photos, so one could see precisely what you are talking about.