Good luck using the Navigator 2.0 API.
My solution is to define the route with isInitialRoute:true.  This prevents Flutter from showing an animation when the route is pushed.
Here's a working example and screen recording:
import 'package:flutter/cupertino.dart'
    show
        CupertinoApp,
        CupertinoButton,
        CupertinoPageRoute,
        CupertinoPageScaffold;
import 'package:flutter/widgets.dart'
    show
        BuildContext,
        Center,
        Column,
        Navigator,
        Route,
        RouteSettings,
        SafeArea,
        Spacer,
        Text,
        runApp,
        Widget;
Widget makeButton(BuildContext context, String routeName) =>
    new CupertinoButton(
      onPressed: () => Navigator.pushReplacementNamed(context, routeName),
      child: Text('Go to \'$routeName\''),
    );
Route generateRoute(RouteSettings settings) {
  switch (settings.name) {
    case 'not-animated':
      return new CupertinoPageRoute(
        settings: RouteSettings(name: settings.name, isInitialRoute: true),
        builder: (context) => CupertinoPageScaffold(
              child: SafeArea(
                child: Center(
                  child: Column(
                    children: [
                      Spacer(),
                      Text('This is \'not-animated\''),
                      makeButton(context, 'animated'),
                      Spacer(),
                    ],
                  ),
                ),
              ),
            ),
      );
    default:
      return null;
  }
}
void main() {
  runApp(
    CupertinoApp(
      onGenerateRoute: generateRoute,
      initialRoute: 'animated',
      routes: {
        'animated': (context) => CupertinoPageScaffold(
              child: SafeArea(
                child: Center(
                  child: Column(
                    children: [
                      Spacer(),
                      Text('This is \'animated\''),
                      makeButton(context, 'not-animated'),
                      Spacer(),
                    ],
                  ),
                ),
              ),
            ),
      },
    ),
  );
}
