He begins with Burke, and it's striking that, after the signal absence of any discussion of liberty in Kirk's checklist of conservative principles, we begin with an excerpt that Kirk titles "The Truth About Civil Liberty." Kirk focuses on "the conservative dislike of extremes," and certainly Burke doesn't endorse absolute liberty -- but then, who, outside Sade, does?
Liberty too must be limited in order to be possessed. The degree of restraint it is impossible in any case to settle precisely. But it ought to be the constant aim of every wise, publick council, to find out by cautious experiments, and rational, cool endeavours, with how little, not how much of this restraint, the community can subsist. For liberty is a good to be improved, and not an evil to be lessened.The obvious inclination here is to a practical maximum of liberty, and Burke's difference with a "liberal," if any, would be about the nature of the "cautious experiments" and "rational, cool endeavors."
Although this extract (from the "Letter to the Sheriffs of Bristol") dates before 1789, it already seems likely that a problem with Burke as a conservative poster-child is that Burke was arguing against what is usually a straw man -- the specter of untrammelled liberty. 1789, arguably, gave this specter flesh and bone; but how much do arguments relevant to the French Revolution bear on ordinary, non-revolutionary politics?
Certainly, when it comes to defining "the real rights of men," there is not much daylight between Burke and J.S. Mill on first principles:
Whatever each man can separately do, without trespassing on others, he has a right to do for himself ....There is of course in the Reflections a great deal about gentle manners and religion, their indispensability for civil society, etc. - but how much of that has been rebutted by experiment and cool rationality?
Burke was probably right that France would have done better to reform its institutions rather than abolish them, and that the absolute power of the assembled radicals led inevitably to terror and bloodshed. (We leave aside the fact that Louis XVI himself did a great deal to oppose reform and to make the radical revolution inevitable; it's a pity that Burke addressed no pamphlet to him.) But I am not sure that a prudent liberal equals a conservative, and prudence is the monopoly of no party.
... Burke's conservatism is better elucidated, I'm sure, by Francis Canavan's introduction to the Liberty Fund edition of the Reflections:
... the right that was fundamentally at issue between Burke and his opponents. They held that every man in the state of nature had a sovereign right to govern himself and for that reason had a right to an equal share in the government of civil society. Burke held that what was important in the civil state was not that every man’s will should be registered in the process of government, but that his real interests (advantages, goods) should be achieved. * * * But this implies that purpose, rather than original rights and individual consent, is the organizing and legitimizing principle of a constitution. * * *This focuses the difference, then, between conservatives and liberals: conservatives believe that there is *a* good for "the mulitude" which can be discerned by the wise. Ultimately, this is a teleological position, which is why Canarvan's exposition of the focus on *purpose* is so relevant. Whereas liberals don't take it as a given that there is a single good for all, or that if there were, anyone would be better able to discern it for the multitude than they themselves.
Who, then, shall make the practical judgments of politics? The question cannot be answered by appealing to the rights of men. “Men have no right to what is not reasonable, and to what is not for their benefit.” But as to what is for their benefit, Burke said: “The will of the many, and their interest, must very often differ.” The first duty of statesmen, indeed, is to “provide for the multitude; because it is the multitude; and is therefore, as such, the first object . . . in all institutions.” But the object is the good of the people, not the performance of their will. The duties of statesmen, in consequence, do not belong by right to those whom the many have chosen, but ought to be performed by those qualified by “virtue and wisdom, actual or presumptive,” for the task of government.
In other words, conservatives seek a first-order state, in which the government is directed towards achieving the good; liberals seek a second-order state, which facilitates the multitude's search for its own plural conception of the good.
Obviously, this ties Burke to metaphysical claims (much tho he would disavow the word) about a moral order in the universe, and makes his reliance on religion and tradition more than mere obscurantism.