> European high-speed rail systems have some dedicated rights of way, but they definitely didn't start that way.
TGV was most definitely started that way. It was engineered to be able to run on regular railway ("standard gauge high-speed train") so that the existing "inner-city" rails could be used to reach inside existing metropolis, but it was designed primarily to run on new specific rights of way, regular service started on the dedicated "LN1" (now LGV Sud-Est) between Paris and Lyon: the line is 409km (254mi) for a total ride distance (station to station) of 425km (264mi).
> They are systems that are integrated with regional and urban rail, and have grown to have more dedicated higher speed sections over time.
The size of the dedicated high-speed network grew over time, but high-speed trains were always intended to run on high-speed networks, there's just no point otherwise, "regular" trainsets will reach older lines's rated speed just fine.