For coincidence experiments at high count rate one may want to preserve a high speed. For just getting the pulse amplitude, there is no need for a super fast amplifier. The pulse shaping is essentially some well behaved band-pass filter. So the is not real need to have the input amplifier faster than the pulse shaper and amplitude analyzer. If the pulse is much shorter than the response time of the amplifier, one would still see a pulse, but a wider one and with reduced peak amplitude, but the same area under the pulse. So a fast amplifier is not absolutely needed. A slow amplifier could be seen as part of the pulse shaping.
For the amplitude analysis there are several options and this also effects the preferred pulse shape. The 3 main methods I know of are:
a) analog peak detector circuit and "slow" ADC (probably the easiest method - I have build this type).
b) fast ADC and numerical amplitude calculation (usually more from area than peak)
c) analog integration with some baseline restoring and slower but higher resolution ADC and looking for "steps" instead of peaks.
The last method does not need an extra peak shaper.
The amplifier should normally be stable enough, so that one does not need an extra marker. The weak point is more the stability of the high voltage that sets the PMT gain. So if at all a test pulse would need to be optical. The alternative is known isotope from time to time.