Part one responds not to the technical difficulties of such a device, but a more fundamental conceptual issue.
Competitive Fencers are some of the fastest humans on Earth (both by self-selection into the sport and the Darwinian aspect that if you are able to win enough bouts to want to keep doing it...) and for us, the difference between you scoring a touch first or me scoring a touch first is set to 1/32 of a second, which is half of the lower end human response time of 1/16 of a second.
And I gotta tell you as a fencer, martial artist and a high performance motorcyclist - that's pretty amazingly quick response time.
So @jdunlop is onto a really solid point here: at the velocities indicated in the OP and the video, it's very unlikely a human swinger-operator could respond quickly enough to not end up pancaked against some built environment element or other.
It's also worth noting that humans "run out" of fast-twitch decisions pretty quickly, reaching "decision fatigue" when they are likely to make wild misjudgements or fail to decide - or more typically just... freeze.
In this scenario, either result is likely to be terminal!
Part two looks briefly at the physics (without maths)
Strength of materials is an immense issue with this concept, both for the device and its components and for the built & natural environment elements being used as hoist points for this idea.
Steel cabling would not be even close to strong enough or have enough longevity under the continually varying stressors applied, and would very rapidly reach metal fatigue and shear. Bear in mind we're not talking about the load of just the mass of the swinger-operator and the mass of their compressed gas canister and gas inside, we are also talking about the inherent load of the length of cabling itself, and more crucially, the force (simplistically Mass x Acceleration [more accurately see the vector equation below])being exerted along that cable (tension) by the moving mass of the swinger-operator and their equipment - and then subsequently massively changed in both amplitude and vector with each new connect-pull-swing-jet-disconnect event. Every hard stop event is a massive deceleration - placing immense strain on those cables - ask a crane operator about working load factors and braking stress - they ease those heavy loads down due to the huge impact sudden changes can have on hoist cables - and in fact the other massive failure mode for hoist cabling is sudden detensioning after overload - leads to a kind of unraveling / unweaving called "birdcaging" which then gets run through the various pulleys and blocks in the crane and the cable gets chewed to heck.
So the only material we know of at the moment which comes close to this level of tensile strength, strength-to-weight and resilience is natural spider silk.
where vector p is the momentum of the system, and vector F is the net (vector sum) force. If a body is in equilibrium, there is zero net force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an unbalanced force acting on an object it will result in the object's momentum changing over time.
Compressed gas has mass - and the more compressed it is, the denser, the higher the mass; in addition, pressure vessels are incredibly dense and have to be manufactured to pretty exacting specs - and they are heavy (ask a scuba diver) in the extreme. This adds immensely to the load on the putative steel cabling - and the more compressed your gas, the denser the pressure vessel has to be to constrain that gas. Moreover, to have enough gas to lift the swinger-operator through space and reverse vectors when necessary means your reservoir is staggeringly large. And of course, you have an ever-escalacting version of the rocketry lemma: more reactant = more mass to move = more reactant needed - but in this case with the additional constraint operating inside a 1G gravity reference frame.
Harness webbing / straps - the harness being worn to throw that human body around would need to be not only strong and tight fitting (with these forces, even the slightest slack will lead to immense impact damage from the straps slapping home on the swinger-operator's body with each event. The straps would need to be hecka wide in any place bearing significant load, and yet somehow conformable enough to body shape to not be incredibly uncomfortable or joint-de-articulating.
Swinger-operator - pretty much either isn't an unaugmented human to begin with or dies very quickly - even assuming we somehow handwavingly beat the reaction time point I raised above, the huge acceleration loads (3-5 gravities) in quick succession in varying directions would induce blackouts, redouts, broken limbs, de-articulated joints, broken spine, broken neck, decapitations...
Built or natural environment elements - each of the things being used as a point of purchase by the "grappling hooks" would be subject to both compressive and tensile loads in very quick succession in very small point-load sized areas, and in most cases we can assume things like concrete, stone, glass and other similar facade system materials - once in a while wood, and even less frequently steel or other metallic surfaces. With these kinds of impacts and loads, the grappling hooks will simply break through materials as load increases; as each grappling hook pops free (punching out concrete fragments from around its attachment point such that they become shrapnel moving towards our swinger-operator) suddenly plunging our swinger-operator out or down into space and shifting not just the remaining but now increased load onto other grappling hook connections - these too will catastrophically fail... I think you get the furry, flat and squidgy meaty pancake on the groundplane idea here.