Why so much throttle work on navy fighter during final approach?
The rapid motions you see are normal, are not excessive in magnitude, and the duration is relatively brief.  The constant corrections are a direct result of the precision required to maintain glideslope, coupled with the sensitivity of the carrier’s Fresnel lens Optical Landing System.  (OLS, the orange “meatball” centered between two rows of green datum lights)

Navy pilots are taught to control the angle of attack with pitch, and glideslope with power.  (he is not trying to accelerate when he adds power, he is slowing his rate of descent)
Deviations from glideslope should be fixed immediately with a three part power correction:

If you are low, add power to drive the ball up until it is just starting to crest high above the datums, then reduce the power sharply to stop further upward travel, then reset power to a new position that should be correct…
If you are high, reduce power to lower the ball until it is centered between the datums, (never allow it to go low!) add a good handful of power to catch it in the middle, then reset power to a new position that should be correct…

The issue that you can never truly find that sweet spot where you can just leave the power set and ride it out.  There are just too many variables, and the ball is very sensitive.  So, one three part power correction leads to another three part power correction, and so on and so forth.  The reality then is what you see; almost constant throttle jockeying in order to control the ball as best you can.
Fine control requires much more rapid corrections.  Think of it this way, if I asked you to ride a bicycle along a painted white stripe 24” wide you could probably do so quite smoothly.  However, if I asked you to strictly follow a line merely the width of the 4&quot; painted stripe along the side of the road, your handlebar movements would be much more rapid as you made corrections to attempt to avoid straying from this thinner line.  Your bike might follow a more precise path, but you would be a bit more jerky in keeping it there.
For comparison, probably one of the more precise optical glideslopes available to civilian pilots is the 4 light PAPI.  (“P” for precision…)  This pales, however, compared to the sensitivity and precision of the Navy’s OLS.
With PAPI the white light fades slowly into red, and even when the colors are distinct the incentive for immediate correction is not as urgent.  If you have three red lights and one white, so what?  You are a little bit low…  Check your VSI, maybe squeak a little more power on, but if you are assured of making the runway your focus is probably on being smooth.  Nobody is watching, nobody really cares that you are a little bit low, and you have more time to correct it.
It is different on a Navy carrier.  You have 15 to 18 seconds from the time you roll wings level until you touch down.  If you are high or low at the start you need to fix it.  You need to fix it NOW.  If you carry that trend for more than a couple seconds people will start talking to you on the radio offering not so friendly “suggestions”.  There is a team of a half dozen Landing Signal Officers (LSOs) standing at the back of the ship watching and recording your every move.  Their first job is to keep you from hitting the ramp, their second job is to get you aboard on the first pass, their third job is to document every deviation (in short hand) for pilot trend analysis and determination of a landing grade that will rank you against your peers.
Video of every approach is broadcast live on closed circuit television all around the ship.  Your squadron commander, your peers, the airboss, captain of the ship, and a host of other random personnel on the ship are all watching you in real time. The LSOs will send you around with a “wave off” if they don’t like how you are responding to deviations, and your landing grades and reputation will suffer.
There is pressure, therefore the incentive to make immediate corrections is strong.  The ball moves noticeably to indicate the slightest deviation from glideslope, so the pilot is going to move the throttle just as rapidly as needed to keep that ball smack dab in the middle where it belongs, and avoid a wave-off or bolter to get aboard the first time.
Worth noting, but this pilot went to full power well before crossing the ramp, so he must have gotten pretty low in close!
Engine spool-up time is not nearly the issue it was in the early days of jet carrier aviation in the 50s and early 60s.  The Navy learned the hard way back then and these days most jets are optimized for acceptable throttle response at approach speeds.
Regarding your final question, the answer is yes.  Navy pilots are even more skilled than Red Bull air race pilots.