Both la Boétie and Edmund Burke point out the role of religion in maintaining the mystique of legitimacy. Religion is an ancient and powerful legitimizing force for statism.
Tyrants themselves have wondered that men could endure the persecution of a single man; they have insisted on using religion for their own protection and, where possible, have borrowed a stray bit of divinity to bolster up their evil ways. If we are to believe the Sybil of Virgil, Salmoneus, in torment for having paraded as Jupiter in order to deceive the populace, now atones in nethermost Hell. ... If such a one, who in his time acted merely through the folly of insolence, is so well received in Hell, I think that those who have used religion as a cloak to hide their vileness will be even more deservedly lodged in the same place. Our own leaders have employed in France certain similar devices, such as toads, fleurs-de-lys, sacred vessels, and standards with flames of gold. - Étienne de la Boétie, The Politics of Obedience
Civil Government borrows a Strength from ecclesiastical; and artificial Laws receive a Sanction from artificial Revelations. The Ideas of Religion and Government are closely connected; and whilst we receive Government as a thing necessary, or even useful to our Well-being, we shall in spite of us draw in, as a necessary, tho' undesirable Consequence, an artificial Religion of some kind or other. To this the Vulgar will always be voluntary Slaves; and even those of a Rank of Understanding superior, will now and then involuntarily feel its Influence. - Edmund Burke, Vindication of Natural Society
La Boétie and Burke wrote as quasi-anarchists, questioning the institution of state but not explicitly opposing it in principle. La Boétie can be construed as only opposing tyrants, certain current players or personnel in ruling roles. But he seems, with his triumvarate classification of states, to include all states in his considerations. If so, he satisfies anti-statist assertion #1, the denial of moral legitimacy, but not the others. Edmund Burke makes an eloquent case that states are even worse than statelessness, but does not come close to saying all states should be abolished immediately. He seems to satisfy assertions #1 and #2, but not #3.
Interestingly, neither of these gentlemen published their quasi-anarchist essays using their own name. La Boétie's "The Politics of Obedience" was distributed privately with due anonymity, and Edmund Burke attributed his "Vindication" to a dead man - the late Lord Bolingbroke, who had been known for radical opinions. Both apparently wanted plausible denial of any anarchist sentiments. Both soon became functionaries of the state and political elites. Both may have been anti-statist in their younger days, but both went quickly over to career statism.