The OP only scratches the surface. It is true that liberty and equality are at odds with each other. This is largely because of the observable fact that God (or nature if you prefer) has not seen fit to distribute His gifts equally. Both physical and mental abilities of all different types range among people from very high to very low, as will what they choose to do with those abilities. To the degree that people have liberty, these natural inequalities will manifest themselves. Equality can never be assured in liberty, only in servitude. History's ultimate manifestation of equality was not in the founding of the United States, but in the Soviet gulag. This is the answer to the question of "how can anyone be truly free if everyone isn't equal?"; it's the only way anyone can be free. Free men are not equal, and equal men are not free.
I am not, however, content with a simple libertarian critique, of equality, because the problem with equality cuts much deeper than that. It isn't simply equality's incompatibility with liberty; it's that any functioning civilization requires a working class structure. History shows that without one a society can rise no higher than a Stone Age hunter-gatherer culture. This is because a functional civilization requires specialization, citizens devoted to performing different roles in its operation, which necessarily entails giving them different amounts of authority to perform their duties, as well as different amount of wealth as compensation for different levels of productivity. Worse still, equality requires suppression of societies most productive elements, such as kulaks in the USSR, or just about anybody who was literate under Pol Pot. How do you equalize an Olympic sprinter and a paraplegic? You can't make the paraplegic a sprinter, but you can make the sprinter a paraplegic! Equality then, must always mean enforced mediocrity.
Finally, because equality can only be enforced by the exercise of power over others, which requires putting some in positions of greater authority than others, pursuit of equality is ultimately pursuit of a mirage. Equality. as a concept, has very limited practical application; it has nearly infinite potential for use by tyrants and demagogues to rouse the masses to support whatever cause they choose to advance. Nietzsche's dictum that the values of the weak prevail when the strong appropriate them as devices of leadership comes into play here. Equality is not simply inimical to liberty; it's inimical to civilization.