Major problem: Context is king in archaeology.
The context of an artifact is its position in the site relative to everything else, and it's incredibly important. A stone flake by itself tells you a little, but a stone flake next to a core by a fire with carbonized bones in it tells you a lot more about the people who lived there. This is why archaeological digs painstakingly record the position and orientation of everything they find, in addition to soil features, etc. that might reveal even more. If you see an artifact in a museum that is from an archaeological dig, there's a book in an archive somewhere that has a drawing of the level in which it was found that gives its orientation and position relative to a datum. You could return to the dig and place the artifact in precisely the same location that it was found in using this data.
Archaeology is an evolving discipline, and better ways of extracting additional information from digs with new methods are being found all the time. For this reason, most major digs do not cover 100% of the site. It's considered good practice to leave some of the site undisturbed so that future archaeologists can apply yet-to-be-invented techniques to the site to reveal more than is presently possible.
So, by scooping up artifacts indiana-jones style, you're destroying a huge part of the information that they might have revealed. Is it better than looters digging the place up? Perhaps, but only just barely. It's a highly destructive, unethical act. It's like justifying a smash-and-grab because there's a riot going on and you happen to be slightly more needy than most. There's a small chance that looters won't find the site or that a government will decide to protect it in time, and it's better to take that chance than it is to loot the place and ensure the site is destroyed.