Does availability of the SRA depend on the make and model of the radar, to any degree ?
Absolutely - as does the processing of the radar data and the display system used.
The principal characteristic of any radar is its wavelength; the shorter the wavelength, the more accurate the picture can be. But with radar, everything is a trade-off. A short wavelength radar will be limited in range and the return is likely to be relatively weak, on the other hand, it will turn quickly and give a rapid refresh to the picture. The Plessey 424 and 430 are both 3 cm radars. At the other end of the scale the classic, or traditional area radar, something like a Marconi 264, is a 50 cm radar - long-range, relatively long time between updates and, at range, blips the size of sausages. In the middle, at 10 cm wavelength, are radars which commonly have been used for aerodrome surveillance. These days, 23 cm radars are quite common because they provide a good compromise between the old really long-range radars and those which are good for terminal areas and aerodromes.
Radars which are used for very accurate surveillance radar approaches need to provide an accurate picture to the controller which is updated frequently. This is traditionally was done using 'raw' data on a classical glowing phosphor tube. In a modern radar the raw radar data is processed and commonly what the controller sees is a position symbol which is where the radar thinks the aircraft is (or is most likely to be). There are a number of approximations that can affect the position of the symbol relative to the actual location of the aircraft - this is why, where SRAs are still available, they are only used for approaches to 2 miles from touchdown or thereabouts. Although I agree the term 'cloudbreak' is rather outmoded, the reality is that at this range and aircraft will be at about 650 feet above the runway and so can be useful to enable aircraft to become visual beneath a layer of cloud in moderately poor conditions.
Now that you've got me thinking about them again it brings back how much fun they were to do and it was very satisfying to do a half mile SRA in poor conditions!