You can bet it will cost tax dollars and private property owners will be affected in ways they never thought of.
Many times projects like this are funded by private grants.
If tax dollars are used they are already allocated for such projects and stimulate spending involved in executing the project. If you have never seen an actual excavation you might be surprised how much hardware and office supplies (sharpies, notebooks, fountain pens with nice nibs not to mention computer needs) are needed. That does not include the food and drink required to keep the crew going nor the transportation costs. Then when you factor in the potential for increasing tourism where none previously existed, the return increases. It is a wise investment.
As far as your implication that somehow someone is going to lose their private property rights, this is unfounded. The only 'loss' I have seen is remorse a land owner had when a nice relic or artifact (that they were unaware of being in the dirt in the first place) is unearthed and removed for display. Landowners who allow access to their property are generally understanding that they are promoting cultural literacy and have no regrets. The paranoid types with the fences and 'posted no trespassing' signs will not grant access anyway, so no need to worry about them.