Your perfectly single-sideband suppressed-carrier modulated sinusoid certainly has a phase which can be measured. However, what you cannot tell is what the contributions of that measured phase from the audio input and the RF oscillator were.
There is another form of single-sideband modulation, in which not only one sideband but also the carrier component is transmitted. This provides a reference which can be used to synchronize the receive LO to the transmit one - normally done to insure exact tuning, but it would also give you the ability to recover the original audio phase.
It is also quite possible, especially with modern DSP gear, to transmit two separate audio channels, one on each side band. This is commonly called independent sideband modulation (ISB).
Many spread spectrum implementations are DSP based and capable of receiving multiple channels at once - GPS being a good example.
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.
Best Answer
Assuming your receiver is a "homodyne architecture (mix TX and RX together)" as stated by mkeith, the two methods (sawtooth vs triangular) will be identical assuming your scene is stationary during the measurement period. The low-pass filtered mixer output will then always be proportional to the range of the scatterers, regardless of a up or down sweep. With a I/Q receiver you should be able to distinguish between a negative and positive beat, but I can't see any benefit. Only generating a up-sweep (sawtooth) will most likely be easier to implement, but that depends entirely on the hardware.
For moving targets, we need to distinguish between 2 cases.
where 'fast' and 'slow' is relative to the sweep-time. For a sufficiently slow moving target, the doppler shift will be negligible and you can approximate it as stationary. You can find the velocity of a slow moving object by comparing the data from multiple sweeps, again the triangular vs sawtooth makes no difference.
I belive the intention of the triangular waveform is that you can now solve the ambiguity caused by a fast-moving object. In a FMCW radar, a moving target may seem indistinguishable from a stationary one. One traditional then introduces the triangular waveform to solve this ambiguity, see e.g. this open access article, especially figure 1.
Note that this only works for a single moving target, when you have multiple moving targets stuff gets more complicated so thread carefully.
In summary: In choosing between the two waveforms, there is a special case with a fast moving object where the triangular waveform may aid in extracting velocity (or should I say: radial relative velocity between the radar and reflector) depending on the velocity and chirp-rate. But for all other cases, the distinction is mute.
I hope that helped, let me know if I should clarify any of the points.