Humans, like crows, are acquisitive by nature. We are attracted to pretty things and are especially intrigued when it's clear to us that other, ancient humans made the pretty things we find -- like the fascinating forms, lines, colors, and textures of potsherds.
In past decades, collecting potsherds - and even far more substantial artifacts - was considered a harmless pastime perhaps similar to collecting postage stamps. Archaeologists have spent more than a century analyzing hundreds of ceramic styles in the US Southwest and the locations where they have been found; so it would be easy to assume that a few potsherds more or less would make little scientific difference.
From an archeological or historical perspective, every potsherd conveys a lot of information about its location - but only if it remains in that location and maintains its relationship not just to the site but to all the other potsherds, artifacts, and structures found there. Archaeologists have developed a system in which researchers can look at a sherd or vessel, recognize its pottery type, know where and when it was made; and from that information determine when people lived at the site, who they may have traded with, or where they may have migrated from. Thus, the more potsherds and other artifacts that remain in place at a given site, the more complete information they will yield and the more accurate will be the archaeologists' scientific assessments of Puebloan culture and our country's heritage.
In contrast, when a casual visitor removes potsherds from their original location, they lose almost all their value as information. They are useless to archaeologists because they have lost all context. They are useless to most persons collecting them except as a temporary amusement. Depending upon where the casual visitor has picked them up, the collection of potsherds may be an act of disrespect to important Puebloan family heritage and ceremonial practices; and may even be illegal. Ultimately, when the casual collector's curiosity fades the potsherds are likely to just be thrown away.
A final argument: Too much has been taken already. Almost all the Mimbres archaeological sites have endured many decades of major looting. None of us need any more artifacts - even the potsherd fragments that looters have left behind. The sheer volume of potsherds and other artifacts already sitting in dark storage in the bowels of museums, let alone in private collections, never to be seen by the public or researched by scholars, is overwhelming.
The GCAS would gently suggest that you follow our group's practice: Find a sherd? (1) Pick it up or (2) leave it in place and (3) photograph it and/or (4) make a sketch of it and when and precisely where you found it. Finally, (5) put it back and (6) walk away, satisfied!