Yes, they fly on handles, and a backstrap is critical in my opinion, I can't imagine using a seat or waist harness with a strop between the handles. This is because cquads need both main line input and brake line input. There is no other kite I've seen that flys the way these do, and accordingly no other kite that is capable of maneuvering they way they do. When properly set up, you can stall the kite just about anywhere and hold it in the sky (I was holding the 12.5 in front of the camera low to the ground). And you can fly them backwards by hitting both brakes hard. Setting the main and brake line attachment points to the handles properly is super important, you WANT to be able to fly the kite backwards using the brakes. If you take off and you can't stall the kite with the brakes, you need to land and shorten the brake lines (or lengthen the main lines).
It is also critical that the lines coming off the back of the handles to connect to the strap are about 1 inch lower than the main lines going to the kite. Without doing this, when the handle strop lines are at the same point on the handles as the main lines, cquads will want to overfly the window. When they overfly, they flop forward and they will fall slowly to the ground nose first (and if you are on hard ground like pavement it will probably break the leading edge spar). There is nothing you can do to stop the crash once the kite has overflown. However, when you have the strop lines lower than the main lines, when the kite attempts to overfly, it automatically pulls its own brake lines and stalls the kite. When perfectly set up, they will not overfly, the will stall themselves just before the zenith of the wind window.
Furthermore... setup of the kite relative to the wind is very procedural. You launch almost directly downwind with the kite on its back lying on the sand, pointed slightly to one side. You pick up the leeward side of the kite by pulling on that handle and the sail will start to catch wind. You can then peel the kite off the ground and fly low to the side of the window and you're good. Often at this moment one of the many bridle lines can snag on a spar tip, etc, and you have to start over, after fixing the snag at the kite. Landing is basically the reverse of this, and is always unassisted.. fly to the lower edge of the window, touch one wingtip to the ground, and then walk towards the kite to lay it down on its back.
In short, these kites are very particular and protocol oriented, unlike modern foils and inflatables, and they have almost no depower so are terrible in gusty conditions. This annoyance is easily outweighed by the power and maneuverability that a good pilot can produce. Easy self landing is also a huge plus. If you are unfamiliar with how to pilot a cquad, these big monsters are probably not the way to start. Better to find a 3.2 or a 4.2 and practice on sand. You'll be amazed at the power.