As you suggest, ancient Athenian democracy bore little resemblance to modern forms of representative democracy and no doubt appears very limited/odd to modern sensibilities. What you wrote doesn’t make what I stated about the derivation of the word any less true though:
https://www.etymonline.com/word/democracy
All forms of democracy so far have been hefty compromises in one way or another and so will all be in the future, short of some sort of technological marvel I suppose. Different countries, constitutions etc. rebalance the mechanisms in one way or another, as more or less direct/representative forms of democracy inevitably have consequent drawbacks... Parliamentary democracies, republican democracies etc./all the various voting mechanisms etc., all are compromises designed to try and strike some sort of balance or confer some practical advantage. You can argue that some are antiquated, some unfit for purpose, but most IMHO. are better than the alternative. The fundamental problem with oligarchies/tyrannies/polyarchy/plutarchies/‘pick-your-archy’ is surely that you might (and almost inevitably eventually will) get a bad apple/apples. ‘Absolute power corrupts absolutely’ etc. As I understand it, the U.S. is a kind of republic (a ‘federal presidential constitutional republic’ according to Wikipedia :) and Madison/Jefferson/the founding ‘dudes’ tried to strike some, as they viewed it, balance between representative democracy and an appointed/elected executive, as you’re no doubt aware… You can argue it was/is deeply flawed and that their efforts to avoid partisan politics/entrenched political parties have failed/there are many problems with lobbying groups/corruption etc., but I would suggest that there exists at least some opportunity to debate and consequently change representatives/legislate a hopefully better way of doing things? It beats the alternative/being arbitrarily ruled by a bonkers king/some tinpot dictator/absolute/hereditary monarch! Perhaps the democratic process ought to always be a work-in-progress, a compromise that we tweak and balance with other mechanisms to eg. protect minority rights or to provide long-term stability and short-term effective and efficient decision-making through discussion rather than dictat? I would suggest that there are decisions best taken at a very direct/local level and others that are best dealt with globally/even on a more technocratic basis… The voters ideally should be educated enough to be able to thoroughly understand the process and enabled and enfranchised wherever possible to engage with it fully…
“Many forms of Government have been tried, and will be tried in this world of sin and woe. No one pretends that democracy is perfect or all-wise. Indeed it has been said that democracy is the worst form of Government except for all those other forms that have been tried from time to time.…”
- from a speech in the Commons by Winston Churchill (a man with many imperfections! :)
https://richardlangworth.com/worst-form-of-government