Your example is right on the money. I used to be consumed by organizational disorder, and even worse, wasn't aware that it was happening. Agile is no magic solution but it can be a vehicle to creating organizational order if you need it, especially if it's a hot buzz word in your company.
Scrum, if you do it right, is precisely the solution. Scrum specifically rejects the idea of "just doing something on the side".
Unfortunately, most organizations believe in cults and magic, and believe (and act) that Scrum is a cargo cult where you say the magic words and hold the magic rituals and get the results.