Was the nonprescision approach into JFK 13L/R designed for smaller aircraft and another era of aviators.
Non precision approaches are certainly from another era. The issue seems to be that there's an increasing number of pilots who cannot fly the visual portion of any approach. If you only ever do automatic landings, then doing one manually seems to be beyond them. There are similar arrivals at many places around the world. Melbourne has one on 34, that comes in over Essendon. It was actually banned for a while after someone got it very wrong...by playing with the buttons, instead of just looking out the window and doing it visually.
Seems to me that for some airlines, flying by using a piece of equipment called the window is more difficult to achieve than operating some dial or button.
I expect that the issue will become worse. The days of pilots learning their business in an air force, or small aircraft, seem to be disappearing, with more and more learning in simulators, with minimal exposure in real aircraft. The Asiana captain who crashed the 777 at SFO was actually worried about having to do a visual approach...
Can automation fly a curved approach like Carnasie or Canasty as some describe it?
I'm sure that it can, but I don't think any of the aircraft are currently set up to do so. The entire approach isn't curved....it's just a turn from Canarsie onto finals. There's still a straight finals leg from about 500' or so. Compare that to the old IGS at HKG, which had you turning right down to about 200', although you only had to turn through 45º.
I suspect that there are many people out there who make multiple waypoints on the curve, so that the system will fly it...to a degree anyway. That's more work than just looking out the window.....
With the “fast speech” from ATC, does the pilot Monitoring write it down to minimise errors?
People often write things down, especially when they expect a long winded clearance. Taxi clearance in particular can be confusing. Writing implies that you have something to write on....which isn't necessarily as readily available as you'd imagine. I've seen guys use the FMC scratchpad on the Boeings for some typed shorthand. The paperless office isn't particularly convenient....