The example of the US Democrats (and pretty much every single western mainstream party which is supposedly left-of-center) doesn't show a Modern Liberalism that matches what you claim is the modern one.

I think you're confusing Left-wing Liberalism with "Modern".

Left-wing Liberalism is basically Social Democracy - trying for Equality of Opportunity rather than trying for Equality of Outcomes like Socialism, totally accepting of private ownership and favoring mechanisms like Public Education and Progressive Taxation rather than more Autoritarian ones to try and make for a more equal world.

Right-wing Liberalism is Neoliberalism - basically "the Power Of Money should be above the Power Of The State", only it's all wrapped up as "the State should not own anything and should not interfere in the Free Market". A nation where Money sits above the State in the power hierarchy can't a Democracy even with a vote and elected representatives since what those elected representatives control - the State - sits below what Money controls, and such power structure is actually known as an Oligarchy.

(If I remember correctly there is actually an MIT study for the US that shows that it is in fact and Oligarchy and has been so for more than a decade)

In the modern days Right-wing Liberalism has been dominant in the West, something you can see even in Scandinavia which were maybe some of the last western nations were social-democracy was still dominant (until maybe a decade ago).

All this to say that in practice the present day (hence, "modern") Liberalism in pretty much all of the West is the right-wing take on Liberalism - Free Market Liberalism, aka Neoliberalism.