The centrality of the king to ancient Egypt is hard (impossible) to overstate – he dominated Egypt on both a political and ideological level, and was essential for life to continue. The king was qualitatively different to regular people, and therefore did things that humans did not. There’s no sense that people outside the elite would have thought that what the king did was weird, or wrong, and there is certainly no chance that they would ever have put these thoughts down into writing: contemporaneous criticism of kingship as an institution does not exist in Egypt.
It's interesting that you’ve asked this question, though, because the traditional popular view of Egypt is that everyone was marrying their siblings, and that incest was rife. This is absolutely not true for Egypt before the Hellenistic period. (Sibling) incest was definitely important for Egyptian myths, particularly regarding the creation of the world (with Shu and Tefnut), and the origins of Egyptian kingship (Isis and Osiris), but this goes back to the nature of kingship: gods were qualitatively different from humans, and so things that the gods did were not possible – because humans were not the gods – for other Egyptians. Robinson’s survey of pharaonic Egypt finds only a handful of cases where sibling incest was practised by non-royals, which over the course of 2000+ years is statistically insignificant, and unlikely to be much greater than any other culture; the same is also true of polygamy, which was widely practised by the king but essentially non-existent for everyone else.
The reason that the king (and only the king) practised both incest and polygamy, in reality and in terms of ideology, is all related to the issue of the king’s gender, and gender in general for ancient Egypt. Returning to mythology, a number of gods associated with kingship had complex relationships with gender: Atum, the creator god, impregnated himself; Osiris, progenitor of kingship, had a penis but was sexually passive in the conception of Horus; Amun, divine analogue for the king particularly in the New Kingdom, could be called both ‘father of fathers’ and ‘mother of mothers’. This is, at least in part, why the Egyptians had no conceptual issue with female kings such as Hatshepsut, Sobekneferu, or Tawosret: divine models for the king had both masculine and feminine attributes. This is particularly pointed in the case of Atum, who conceived the world by creating intercourse between a grammatically masculine part of his body (his penis) and a grammatically feminine part (his hand); indeed, in some artwork we find a goddess named ‘His Hand’ accompanying Atum.
Obviously, Egypt was still a structurally misogynistic and sexist society that favoured men, which is why female humans only rarely became king (at which point they became both masculine and feminine simultaneously). Kings were normally male, and kingship was occupied by a male body, but the feminine aspects of the king were still present and had their own duties to deal with (particularly interacting with female deities such as Hathor). As a solution to this, the feminine aspects of the king were essentially transmuted into female bodies, the female relatives of the king. Any royal woman had as her title something along the lines of ‘king’s mother’, ‘king’s sister’, ‘king’s wife’ etc.; they’re all conceived of not as individuals (as English ‘queen’ implies), but as simultaneously possessions and extensions of the king. So the king’s sisters were essentially part of the king himself, and marrying them and creating new life with them was conceptually the same as Atum creating new life with his hand.
Now, whether any Egyptian king actually thought about why they were doing this is hard to say – it was traditional, and the divine model of Osiris and Isis was highly compelling: the king was conceptually Horus, their son, but if the king’s own son was also going to be Horus, then the king also had to be Osiris, which meant his wife had to be his sister Isis, and so on and so on. There are also strong political reasons: in the same way that you wouldn’t want to transfer any of the king’s divine power outside of the family by allowing a non-royal to marry a king’s sister, you also wouldn’t want to transfer political power out of the family that way either; in all the marriage diplomacy going on during the New Kingdom between the great powers of Egypt, Mesopotamia, and Asia Minor, the Egyptians are markedly more resistant to allowing any of their royal women to be married to foreign kings.
As for your standard, everyday, regular Egyptian? I don’t think he would have thought about this at all, and a poem by Alexander Pope comes to mind: ‘Know then thyself, presume not God to scan; / The proper study of mankind is man’. And whatever the Egyptian king was, to the Egyptians he was certainly not a man.
Secondary sources
Allen, J. P. (1988), Genesis in Egypt: The Philosophy of Ancient Egyptian Creation Accounts (New Haven, Conn.)
Almansa-Villatoro, M. V. (2020), ‘The Gender Ambiguity of Fertilization: The Hemusets as a Case Study’, ZÄS 147: 9-18
Cooney, K. M. (2010), ‘Gender Transformation in Death: A Case Study of Coffins from Ramesside Period Egypt’, Near Eastern Archaeology 73: 224-37
Manniche, L. (2002), ‘Goddess and Woman in Ancient Egypt’, JSSEA 29: 1-8
Menu, B. (2001), ‘Le mariage en Égypte ancienne’, ÉAO 20: 17-24
Orriols-Llonch, M. (2015), ‘Semen Ingestion and Oral Sex in Ancient Egyptian Texts’, in Kousoulis, P. & Lazaridis, N. (eds.) (2015), Proceedings of the tenth International Congress of Egyptologists : University of the Aegean, Rhodes, 22-29 May 2008 (Leuven): 839-48
Robins, G. (1986), ‘The role of the royal family in the 18th Dynasty up to the reign of Amenhotpe III: 1.Queens’, Wepwawet 2: 10-14
Robinson, J.-M. (2020), “Blood is Thicker than Water”: Non-Royal Consanguineous Marriage in Ancient Egypt: An Exploration of Economic and Biological Outcomes (Oxford)
Troy, L. (1986), Patterns of Queenship in Ancient Egyptian Myth and History (Uppsala)
Wilfong, T. G. (2010), ‘Gender in Ancient Egypt’, in Wendrich, W. (ed.) (2010), Egyptian Archaeology (Malden, MA.): 164-79