> Modern fire trucks, and police cars usually, are built to be able to push vehicles out of the way.

Not so much police cars, anymore.

Back in the time of the B-Body Caprice and the Crown Vic, sure. These days with the exception of the Tahoe the most common police vehicles are all unibody platforms. Charger, Durango, Explorer, Taurus, and the rare Australian Caprice

You can still bolt a push bumper to them and most departments do, but they have to be used with a lot more caution and a lot less aggression to avoid damaging the vehicle than in the days of body-on-frame sedans.

Fire trucks on the other hand, yeah they're basically the opposite in that there might be a couple of Explorers or Durangos in the fleet but most everything else is a medium duty truck or a custom chassis specifically for fire service.