I'm a Gyroplane CFI and former owner/operator of a J-2, with extensive experience in flying the productions models, including transcontinental U.S. trips. I'd like to clarify some of the comments made here, because the aircraft hasn't been given a fair shake. It is fun to fly with no handling vices.
First, the zero climb rate speed mentioned above is a logical necessity in any gyroplane. As one slows, in an aircraft that can neither stall nor hover, there must be a minimum level flight speed. In the J-2, that's around 25 IAS. If you go slower, you will sink, and if you wish to climb, you will need more speed.
This is perfectly normal and not at all dangerous. The downside is that engine cooling airflow isn't great in this condition, so one shouldn't overly prolong it. You will never come anywhere near this condition in any competently flown landing, generally done power off. You can sink vertically safely at zero airspeed, but the sink rate is too high to carry to touchdown, so that isn't a practical landing option.
Next, I never experienced any difficulties in low altitude handling, and indeed would prefer to be in a J-2 suffering an engine failure than in any airplane or helicopter you might name. The aircraft climbs, cruises, and lands all in autorotation, and no pilot action is required to enter that mode of flight or to control rotor rpm. Just point it at your landing spot, and do a cyclic flare at the bottom. You can land it in a tennis court.
Third, in the production models I have flown, the pitot presented no problems and airspeed indications were reliable. The production models, compared to the prototypes, had a large fin between the mast and the prop, and no wheel pants, and showed no yaw instability. Many were fitted with a three blade controllable prop and benefited from a higher gross weight.
Fourth, I have watched the video linked above and suspect that the pilot was not qualified/trained for the unique qualities of the aircraft. I never had a take-off roll of more than 200 feet and routinely landed with zero ground speed. The climb rate is not impressive, but the aircraft is nimble and easy to fly with light forces in all directions on the cyclic.
Fifth, long roll-outs are just plain silly in this aircraft, and there is no reason to do them. If you want to stop far down the runway, you simply fly it there, and then do a minimal roll touchdown. Even on a calm day, you should never need more than the width of a taxiway intersection to touch and stop.
Sixth, the crosswind technique described above smacks of serious incompetence. If there is enough wind to cause concern for a crosswind component, there is enough wind to land with zero roll by merely pointing the nose into the wind. A typical approach is flown with closed throttle and moderate airspeed (I used 53 solo and 57 with a passenger) down a fairly steep profile, One continuously applied cyclic flare would easily bring you to zero ground speed just a couple inches above ground, followed by a gentle settling onto the mains. The nosewheel is then lowered, and you're done. You will have to apply throttle to gain taxi speed.
I preferred to pick a spot where a taxiway intersects the runway and put it down there for an easy exit. Learning that cyclic flare is no harder than learning to flare a taildragger or to quick-stop a helicopter, and doesn't require special abilities or skills. It is merely a new skill for those new to the aircraft.
By the way, the article cited above (airfactsjournal) is replete with bad information and a pretty significant negative bias.
Finally, the question " Why have a machine that is not a helicopter if you can have the real thing? " makes a huge assumption that one can as easily have a helicopter. The J-2 is dirt cheap by comparison to acquire, much simpler and cheaper to maintain, and much less workload to fly. It won't hover, but most helicopters spend precious little of their lives actually hovering for anything but take-off/landing. They are not efficient, but they are great fun if flown well. Unfortunately, they are also sufficiently forgiving to allow clueless pilots to get away with horrible technique, and that's where most negative comments arise.