No, you don’t have to take a backpack. You can follow a backpacker travel style with a suitcase. However, keep in mind that once you leave the airport there are very few occasions where you’ll get to wheel your suitcase for any distance. Cobblestones aren’t luggage friendly and neither are the steps in most hostels, many hotels, a fair number of train stations and most subway stations. Many public transit systems (RER and metro in Paris come to mind) are a nightmare with any luggage and a pack is at least a tad easier getting through turnstyles and standing on the subway. Wheeled luggage makes it easy to overpack and the extra weight of the wheel assembly adds to the difficulty of hoisting your bag onto the train and into the overhead luggage rack.

That said, I often travel with a small (like a pilots case) wheeled suitcase that I can easily carry up steps and a small backpack which I use for my carry-on and use as a day pack or for 1 or 2 day side trips when I can leave the suitcase behind.

Try the Oldlady test. Pack up your gear. Walk around the block twice (preferably a block without curb cuts and with sidewalks rough enough to make wheeling difficult), then immediately walk up 2 flights of stairs — rough equivalency to getting up the subway steps, walking to your hostel and getting to the check-in desk in most cities. If your suitcase feels fine, then forget the backpack.