It's TV, It Ain't Cinema
(1) The style isn’t original or unique. It’s a familiar approach to filmmaking on display in many theatrical releases and in plenty of television. The run-and-gun look is one option within today’s dominant Hollywood style, intensified continuity.
(2) The style achieves its effect through particular techniques, chiefly camerawork, editing, and sound.
(3) The style isn’t best justified as being a reflection of Jason Bourne’s momentary mental states (desperation, panic) or his longer-term mental state (amnesia).
(4) In this case the style achieves a visceral impact, but at the cost of coherence and spatial orientation. It may also serve to hide plot holes and make preposterous stunts seem less so.
I agree with all of these but I thought I might add a couple of points about the origins of this style and possibly why it is more prevalent now than in the past.
We can summarise the style as comprising loosely hand-held camera work, sharp focus pulls and then blistering quick edits in the cutting room. I first recall seeing this style of film-making on television in the mid-nineties. I remember people at the art-college I attended getting very excited by the series "This Life" which featured the hand-held camera and the hard focus-pulls if not the quick edits. But this was television, and a TV screen is contained comfortably within your field of view, is inherently less immersive and requires the use of big graphic styles to make any impact. We can, and often do, move around whilst watching TV so our spatial relationship with the screen is quite dynamic. This is not so at the cinema. More kinetic styles work well on TV because it is less intrusive into our field of view and its smaller scale means it has to be bolder to be seen - it cannot perform subtlety of style like the cinema can.
The style was simultaneously being used by documentary makers, often those on a budget and shooting with just a single camera. The quick reframes and focus pulls would be edited out were more coverage available but the lower the budget, the less coverage and so more of these "mistakes" end up on screen and eventually become indicative of a small-scale, non-corporate project. Even now documentaries can be seen with this style. "Mondovino" by Jonathan Nossiter was self-funded and self-shot and though its subject matter, the international wine industry, is rather more sedate than a Jason Bourne outing Nossiter's film has the hand-held, pull focus style that we come to associate with the rebel documentary maker.
These are tropes for mainstream Hollywood though. These films are not low-budget, nor are they really run and gun film-making. The VFX studio Double Negative has put months of work in on Greengrass's film, making all that expensively shot material look as though it were captured just as it is seen. The use of green screen elements, CGI are all on a par with any other big budget action film, because that's exactly what the film is. Hollywood hopes to get us to believe that this is a "grittier" more "real" vision of the action film by aping the mannerisms of the indie documentary or edgy TV series. I don't agree. It makes it seem like lazy film-making for television, not cinema. When one then considers the hack and slash editing style that is then layered on top to ramp up the kineticism further we often end up with an incoherent mess.
It is the pop video editing piled on top of the rebel documentary visual stylings that finally push them into the absurd. The relentless manner of these films' construction in fact makes the drama and narrative terribly one dimensional. If there is no variance in pace, framing or movement then the eye and mind adjust to it until the fast cutting seems normal. Speed is relative. If there is no slowness to contrast the frenetic action then action seems staid. Three rapid cuts after several longer ones have impact. Three rapid cuts after 120 previous rapid cuts have none. By shooting and cutting dialogue scenes like a low budget pop promo doesn't make the film more exciting overall, it in fact reduces the impact of the action when it begins.
I often wonder whether a great deal of the televisualising of cinema has come about because of the reliance of directors on video-assist technology. These devices allow the director to watch exactly what's being filmed on a small TV set as it is being shot and then use that video to edit the sequence before the film cans have even had the chance to be taken for development. The days of 'dailies sessions', where rushes of the previous day's film are screened on a cinema screen, are gone to great extent. Directors and editors do much their work on a film by looking at a TV screen or computer monitor. I suppose we shouldn't be all that surprised that they end up making faux indie documentary television not cinema.