Yes, the market exists. But it's an illegal market, and the penalties can be, depending on where you are, quite severe.
Adopting that stance, though, is not unlike saying well, murder already exists, why not make it legal and make some money off it?
"IF legitimate archaeologists/institutions/museums sold their "useless"* artifacts, how does that encourage illicit trade that already exists? I would think it would lessen the demand for illicit artifacts."
It undercuts the laws that exists to prevent such illegal trade.
"Also, how does a "for profit" excavator destroy all the value? The whole POINT of these artifacts is their historical significance."
Read what I wrote again. The value of these artifacts is FAR greater than just their monetary value, as I pointed out. For profit excavators, interested in ONLY the monetary value of an object, destroy their contextual value.
"Someone is going to pay big $$ for a clay pot some guy claims he found in the ground in Virginia? Really?"
I don't know what the market value is for Colono-ware pottery in Virginia, but I do know that the illegal market value for pre-Columbian Navajo, Hopi, and Anasazi pottery can be... and it can be staggeringly high.