Others might be able to tell you about the ancient and early medieval history of the Sphinx. But for the interim period until the structures were excavated by modern expeditions, the short answer is that stopping things in the desert from being buried, or excavating them later, would be a huge expenditure, and nobody expected to find anything of value that would justify doing so.
People did indeed go out to the pyramids and explore them. The Ottoman travel writer Evliya Çelebi (1611-1682) says that while in Egypt:
I spent five or ten days partying next to the pyramids, while our horses were grazing in the nearby meadow, and I had occasion to tour them several times and marvel at them.
Though he doesn't mention the sphinx specifically except that the smallest pyramid is known for it, using the local Arabic name, "Abu al-hawl." (Literally 'the father of dread'). He says he and his companions made some expeditions into one of the pyramids to look for treasure or artifacts but only saw skeletons and mummies.
Hundreds of years earlier the Abbasid Caliph Ma'mun (786-833) supposedly made an organized expedition to breach the great pyramid and find out what was inside - not necessarily to look for treasure, but maybe to find ancient texts - 'intellectual treasure', perhaps. Whether or not this actually happened is questionable because some contemporary sources don't mention the expedition, and the main source for it is the Egyptian historian al-Maqrizi (d. 1442), but Robert Cooperson argues pretty convincingly that the expedition did take place. Another Egyptian historian, Muhammad ibn Abdallah al-Idrisi (d. 1251) relates, based on his reading of several earlier Egyptian accounts, that the Caliph did indeed enter the Pyramid:
When the caliph al-Ma'mun — who was interested in exotic knowledge and occult sciences and commissioned the translation of philosophical works from Greek into Arabic — came to Egypt and saw the pyramids, and could find no one able to satisfy his curiosity about them, he ordered the breaching of the Great Pyramid so that he might view the secrets long hidden within it.
But according to these historians the Caliph, if he did really breach the pyramid, didn't find any treasures within. al-Idrisi relates - with some skepticism - a legend that the expedition found only a vessel containing 1,000 gold dinars. The Caliph then ordered that the expenditure of the expedition thus far be tallied up, and the amount spookily came to exactly 1,000 dinars. The Caliph then remarked on the uncanny ability of the pyramid-builders to predict the cost of entering the pyramid and ended the expedition.
Evliya Çelebi recounts this same legend after narrating his own pyramid exploration. So it seems that while people were certainly interested in going to and exploring the pyramids, they wouldn't have thought to make a huge effort to excavate the sphinx or any other structures, because they were aware that the pyramids had already been raided for valuables. Furthermore, there's the idea - still captivating even in modern times - that if there were hidden or lost treasures, the Ancient Egyptians had some uncanny or supernatural way of protecting them. Çelebi says:
Thus this humble one saw the inside of one of the pyramids. There was no buried treasure, only buried men. But there is no doubt that it is under a talismanic power. For when we came to that pool, all of us were confounded, and we remained in this stunned condition until we came out into the air, when we recovered our souls. God save us! May I never be vouchsafed to enter it again.
So there is this idea that maybe there's still something to be found under the pyramids, but it's probably protected by some supernatural power that would prevent it from being found.
Dankoff, Robert. An Ottoman Traveller: Selections from the Book of Travels by Evliya Çelebi. London: Eland, 2010. Kindle Edition.
Cooperson, Michael. "Al-Ma'mün, the Pyramids, and the Hieroglyphs," in John Nawas, ed. 'Abbasid Studies IL Occasional Papers of the School of Abbasid Studies Leuven, 28 June 1 July, 2004, Orientalia Lovaniensia Analecta Series 177 (Leuven: Peeters, 2009)