For one thing, it doesn't appear as if all parts of it were recovered:
Slowly, she reached down for a bag at her feet, turned it upside down, and tipped a dozen bits of splintered wood and twig onto the bed, the only remains of Harry's faithful, finally beaten broomstick.
Harry Potter and the Prisoner of Azkaban, chapter 9
The way this is presented - that's it's all that remains - implies that it isn't all the pieces of it. It was smashed to bits, which means that it would be almost impossible to recover all parts of it. It wouldn't do much good to only put half the broomstick back together!
Additionally, magic has limits. The broomstick was really destroyed. It's possible that magic couldn't fix it.