Pain management is important. Three weeks is a very long time if he no longer needed it after week one. If he actually needed it past week one, I would think he would have also needed other medical attention. If it is a daily dose that is appropriate for his weight and is broken into two administrations, that makes sense. It is chronic use that is of concern w bute over extended periods of time - but I have never seen what the period is, exactly. It is effective as a pain killer, so for short periods of time, it is the "go to".
My sister had a retired stud. As he got older, he had got more severe DJ, so as a quality of life issue in light of his then age (27), he was started on bute daily. He had very happy and healthy twighlight years, and died at 34.
I would not ever give any horse bute for more than a week unless I knew exactly why I was doing that (i.e., I had a clear vet report of the injury, etc.,.) with some direction (e.g., it will take at least a 2 weeks for the condition to improve, in which it is important to keep the pain in check).