Yes. Arrows were really expensive, especially once you get to specialized war arrowheads, barrel-shaped shafts and proper, professional fletching. Getting some of them back is always a good idea. However, there are a few problems with this that limited the use:

1. You don't have all the time in the world to collect them. Your enemy is still somewhere near, regrouping, so you can at best grab what you can for a few minutes and be off to do more proactive things, like fortifying your camp, or marching to besiege a castle.

2. Arrows break. Especially if they hit something hard, like armor or shield, or just a rock. Let's not forget that they were fired into a crowd of people, so they'd be likely to be trampled on and broken that way. It's hard to say how many would be broken this way, but I think it's safe to say much more than a half.

3. Arrowheads could be blunted or bent and need sharpening or reshaping. Depending on how good your supply lines are, this may not be worth the effort to fix.

4. Soldiers are lazy bastards, just like all of us. You may not look all that hard when ordered to look for arrows by your lord and just lazily walk around plucking a few easy finds, focusing more on the more profitable (to you, that is) loot.

5. Some arrows are embedded in people (sometimes still alive people) and that is plain icky to collect. Hell, not everyone has stomach to just walk over a field of very recent battle. By the way, PTSD among the soldiers and knights was absolutely a thing, and there are even some accounts of deeply traumatized knights, nightmares and being triggered by clanging metal and all.

Despite all of this, it was still done, especially during sieges. A funny example is battle of Red Cliff from Chinese history, where they supposedly used straw dummies of soldiers to bait enemy fire and re-stock that way (done on a foggy night) - supposed profit was 100 000 arrows.