Liberty promotes equality in the sense that it recognizes human nature and it recognizes that individuals should be treated, justly under the law or in society generally.
Conservatives seek to conserve the traditions and ideas man has learned to be good through the centuries. Radicals seek change for change's sake.
The ideas of Babbitt, More, and Santayana had small circles of following but didn’t capture the general public’s attention. The nation greatly changed in character and temperance from WW1 to WW2 and conservatism was adrift.
George Santayana had a rich depth of conservative ideas that identified and fought against the materialism of the modern world.
Paul Elmer More is thought of "as a critic of ideas, perhaps there has not been his peer in England or American since Coleridge". He fought the pragmatism of James, the naturalism of Dewey, the sentimentality of the socialists, and the presumption of those that forgot dualism.
Irving Babbitt saw that man was reduced from a state of the exercise of the Will (bound by Rule of Law) to a “gross materialism” and a “single sensate level”. Man destroys himself if he stifles the power of the Will for the anarchy of beasts (Rule of Senses). In him American conservatism reaches maturity.
Conservatism was in a sort of "wilderness" and struggled through the beginning years of the 20th century. Great leaders of ideas were hard to come by, and it seemed that the vision of America espoused by Hamilton had won. Ideas shifted from the political leaders to the educational leaders.
Conservatism is not averse to change, but if there is an overall benefit to society or humankind, a change is worth implementing. If the change attacks the "permanent things", breaks down societal order or throws humanity into chaos, the change is to be resisted with discussion and ideas.
Man is a creature with a dual nature: an individual spirit that longs for a sense of connection to their fellow man. Conservatives celebrate that individual spirit but also seek to nurture a voluntary community built around shared principles, experiences, and a sense of something greater than the individual. The Radical seeks to destroy these communities at every chance they get. When the individual is isolated they are ripe to be exploited by the Radical's siren call of "community" which is nothing more than a despotic state of misery.
The conservative is ultimately concerned with just what effect a new idea or policy will have on the societal order. Radicals are so insistent on change for the sake of change that they cannot comprehend what their ideas will ultimately do to the bonds that connect generation to generation.