The only flaw in the "press on to the full base maintenance and support facilities" approach is that you don't know exactly what the damage is. The shattered spinner debris had potential for causing secondary damage, the vibration before shut down the same, was it more than one bird? Was it a bird? Only by landing and having some level of engineering inspection can you absolutely rule out secondary or additional damage.
In this case, descending to Birmingham was arguably as expeditious as peeling off to an en route alternative, but it won't always be the case. A good captain's decision will have more than a little curiosity about exactly what has happened to the aircraft and a desire to secure that knowledge sooner rather than later.