HSR does indeed need a route past slow trains to reach city centres - but this has now been solved in loads of ways in the 12,000 miles of new lines built in the last 2 decades all over the world.
In most places there is an existing rail route that can be expanded with extra tracks, or a freeway that could have 2 tracks laid over it on a viaduct (as with New York's skytrain). More typically the route is put into a tunnel for the final section to the city centre (as our HS1 line runs into London via the Stratford London Olympic site that you are about to see on TV). The tunnelled route is the most expensive - but have no planning problems!
As for stations, a HSR will have very few. Your mass transit systems are supposed to get people to those HSR stations - which may be SOLELY in the city centre itself and just outside at a single park & ride or airport hub interchange location (eg Frankfurt city centre and Frankfurt airport or the central Paris stations paired with Paris CDG).
In Europe, trains have taken the place of many short haul air journeys and airlines now own some train companies. Air France recently announced that it wanted to run trains instead of planes