What power limitations was chirped radar designed to overcome?

I'm not a radar expert by any means, but I think I understand the general concepts well enough to try to answer your questions.

What specific requirements on the peak and average powers and the widths of radar pulses was chirped-radar designed to overcome? Were these purely 'internal' concerns regarding the electronics, or were there external goals and restrictions that were hard to meet otherwise?

The basic problem in radar is to get both adequate power for total range and good timing resolution for range resolution. It is hard to build high-power amplifiers for microwave frequencies. You want to have a lot of energy in each transmitted pulse, but you also want to keep the pulse short. The solution, as you have found in optics, is to stretch the pulse by chirping it, which allows the power amplifier to operate at a lower power for a longer time in order to get the same pulse energy.

Now, in radar, it doesn't matter if you don't compress the pulse again before feeding it to the antenna — the chirped pulse works just as well as the compressed pulse in terms of detecting objects.

In fact, you gain additional advantages when the reflections come back, because now you can amplify the chirped signal in the receiver (getting some of the same advantages as in the transmitter amplifier regarding peak-to-average power), and you can use a "matched filter" to compress the pulse just prior to detection, which has the additional advantage of rejecting a lot of potential interference sources as well. The narrow pulses coming out of the receiver filter give you the time resolution you need.

Is the name 'chirped pulse amplification' ever used in a radar context?

Generally not, because amplification isn't the only reason that chirping is used.

Is the optics-style CPA - stretch, amplify, compress, and then use the pulse - used at all in radar applications, or in broader electronics fields?

Not to my knowledge, but it would certainly be feasible.

The tactical requirement Cook is talking about are reliable target detection in noise and jamming, this is the problem of detection, and reliable target resolution against coherent background, this is the problem of discrimination.

In a conventional pulse radar these two problems are solved by increased pulse energy and reduced pulse width. The shorter pulse has better chance to be seen by itself than a longer one when multiple targets are present simultaneously and since the matched filter output signal-to-noise ratio is independent of the pulse shape and is maximum among all possible noise filters the tactical problem is solved by having a radar signal such that its matched filter has length that is as short as possible so multiple target returns are well separated in time. So for the radar performance what matters is not what the radar pulse is but what happens after the echoed pulse comes out of its matched filter. Since the matched filter's output amplitude, and hence its SNR, is proportional to the transmitted pulse energy we can manipulate, modulate, what we transmit and achieve the same tactical performance as long as the received SNR and post matched filter pulse length is the same.

Since the performance depends on transmit energy and is independent of transmit power, and all radar transmitters are power limited, radar designers never intentionally use amplitude modulation and all intra-pulse modulation is either phase or frequency. A typical and oldest in a conventional pulse radar is chirp radar but there are many other frequency or phase modulation schemes. While chirp is the oldest and conceptually the simplest, for very sensitive radars it is rarely used. The reason for that is that the output of the matched filter for a chirp radar generates an output away (so-called time sidelobes) from its desired peak that is higher in amplitude and longer in time (ringing) than sometimes is desirable. This high level "ringing" prevents discriminating smaller targets by the output of the a larger target that are near to it. Also the chirp signal is more difficult to process than some others when there are several moving targets, etc.