The monument already has been tidied up, and fairly recently at that: in the spring of 1958, as part of the last major set of excavations there. Mainly a matter of stones being re-erected.
There are also cases like Newgrange, one of the neolithic tombs overlooking the Boyne valley and possibly the best known archeaological site in Ireland. It was originally a low earthen mound, with some sort of stone facade and a tunnel made up of large stones leading to a central chamber. Over time the mound spread and the facade was scattered. Between 1962 and 1975, a team led by Michael O’Kelly excavated the mound to the extent that they completely exposed the passageway, which they then dismantled. On finishing, the monument was restored by rebuilding the passageway, covering it with a layer of concrete to mechanically protect it and then reheaping the mound. The controversial aspect was their rebuilding of the facade. In effect, O’Kelly did put the equivalent of the limestone back on. It’s certainly impressive, but many people also feel it’s too steep to accurately reflect what the builders would have done. Basically a wall that steep with earth behind it, without the modern internal strengthening the reconstruction has, would have fallen over in decades. Of course, the original facade did fall over, but it’s at best uncertain whether it happened that quickly or over thousands of years. Whatever the truth, what tourists see today is at least partly O’Kelly’s best guess. Though a guess defended by him in his (recommended) book on the excavations and explained by the guides showing visitors round.
Bottom line ? Historic sites have changed to greater degrees than people seem to expect. Attitudes change and a policy of nonintervention in one time and place may not translate to another site at a later date.