Yes a root cause inpection as well as a full system inspection after a serpentine belt failure is an excellent idea. After my 120k (110k actual) maintenance I had my belt replaced but not the tensioner.
At about 125k, the plastic in tensioner wheel disentigrated in the middle and that cause the belt to get frayed and whip around the engine.
That's why I'm using this to avoid this problem.
Note that it's metal, not plastic compared to the OEM part.
-back to why inspection is important...
The root casue was the tensioner wheel, but I replaced the idler pulley just because that's also a known failure part.
About 2 days and 50 miles from fixing the belt and tensioner, I get a P0410 error (check engine light). After popping the hood and looking for the culprit. I found that the frayed belt had whipped and damged the hoses that conected to the secondary air pump.
That's why root casue and full system inspections are necessary. Otherwise you might be treating the sysptom, not the casue.