Don't agree? Just look at how the red half of the US talk about Obama and the Blue half talk about Trump.
Maybe look at Ireland or Portugal as examples of republics where the president keeps the constitutional safeguards (dissolving parliament, vetoing laws, and what not), and you'll see that they are not exactly polarizing figures. Instead, the prime ministers are, as they should (they head partisan politics).
I also don't care about how dutiful the Queen is and what she's sacrificed to the role - though to be clear I do believe both those things at a personal level.
It doesn't matter what incidental good it accomplishes if it goes against the fundamental principles of equality.
Even going back to only 100 years ago the idea that "we are all equal" would have sounded ridiculous. As an example, we could take the accounts about the role of schools in various educational systems in Europe: it was widespread belief that the function of such institutions was to select the ἄριστοι - àristoi, from which it derives aristocracy - the best among the best. School was intentionally set up as an exclusionary institution, opposed to our "inclusive" endeavor: it was considered obvious, a "fundamental principle" even, that some people were better, in some sense or in another, than others.
Fundamental means foundational. But regardless of whether or not the definition you choose to follow (and they do vary) adds more nuance, it's pretty obvious what was meant by the phrase fundamental principle. And pretty obvious from the context that it was in fact referring to, as you put it, the "modern conception".
It really feels like you've picked up on a trivial phrasing at the cost of ignoring the intent, probably as a means of introducing some eloquent but irrelevant knowledge on etymology and derivation which adds nothing to the discussion.
It doesn't matter what it was like 100 years ago. We live now. And equality is fundamental to our current conceptions of an ideal democracy. Or foundational if you're still offended by the word choice.
From the Proclamation of the Irish Republic, 106 years ago.
> Even going back to only 100 years ago the idea that "we are all equal" would have sounded ridiculous.