But, this is FR—the land of great debate and controversy—LOL.
First of all, your post points out that Principles, not simple formulas are the bedrock of true political integrity. If conservatives in general could agree on that their would still be a lot of common ground. I don’t know if you read the Nisbet article that I linked early in the thread, but he goes a long way in pointing out the common ground while not ignoring the differences.
Different views of Rights are indeed much of the difference. I found a wonderful explanation in both Thomas Sowell and in Burke. It deals somewhat in what Kirk called the Principle of Prescription. Prescription deals with things like property that are held, not from logic, metaphysical rights arguements or even law, but held from time and honor. The point that Sowell makes in A Conflict of Visions is that those holding with with Constrained view of mankind (Hayek's believers in the Whigish English classical liberalism) believe in Rights that they actually hold, or can hold. He contrasts that with the Unconstrained view holders who believe in rationalist logical constructs that can be debated away, argued away, prioritized away and utilitarian measured and put one right against another.
Burke points out that the Glorius Revolution insured Rights as Englishmen that they already held, not rights that they were due and which then could be legislated or argued away.
Like yourself, Hayek doesn't like the term liberal, or conservative. He also finds Libertarian somewhat lacking. He settles for "Old Whig" saying it is the only thing accurate enough.
I see libertarians who throw away the vagueries of Principles and want the cold logic of "one simple single formula" and while there are some in their rank I respect, and even enjoy, that system is not one I can admire. It sets them on the Rationalist Totalitarian line that Hayek condemns so heartedly. He even alludes to that exact simplistic failure directly when he brushes aside "the younger Mill"
I, quite frankly, don't feel that conservative implies that conservatives are largely defined as wanting to "conserve" anything as the central part of their thinking. The root was used to imply that they weren't ready to abandon civilized society and join the Jocobins. That's the total indictment.
Conservatives, in this day and age, have a radical agenda. And the root of radical is "root" as I recall.