Artifacts are better preserved in the ground for future generations, since technically, archaeology is a destructive process. Besides, archaeological research should be conducted to meet specific research goals unless a site is threatened with destruction. It is dangerous to dig in old wells. The sides of the well could collapse and you could be hurt or trapped. If the artifacts were on private property, they belong to the property owner and it would be stealing if you took them.
Most artifacts are valuable in an historic sense only.
If you removed them, it destroys the chance to learn more about the past.