A fairly simple and physical thing at work, I suggest ..
If the minimum level is going to occur
(a) over the runway, it makes sense that it be radalt based, particularly as one prefers a higher accuracy as the distance to the hard bits below decreases. However, one needs to be over a known ground datum for the thing to make repeatable sense. In the case of cat 2/3, where the aircraft will be over the runway or immediate runway end environment and the position at minimum is known with a reasonable accuracy, it works fine. Given the aim of a cat 2/3 approach is to push the visual environment to critical limits, using a baro decision would make little sense, given altimeter errors.
(b) somewhat prior to the runway end, local terrain is going to complicate procedures sufficient to the extent that a radalt alert has not much meaning other than for general terrain proximity warning where that might be appropriate. Hence the baro decision for minimum is appropriate. While a cat 1 ILS is a precision approach by definition, generally it terminates somewhat prior to the runway end with the necessary need for baro rather than radalt decision.
Procedurally, for (b), some operators get rid of the DH alert altogether, others use it as a backup to the baro decision provided that the terrain environment makes that philosophy sensibly workable - otherwise it is wound out of the scenario.
I was brought up on this latter philosophy and thought that it worked fine .. it always seemed sensible to me to use whatever warnings and alerts were useful to maximise the likelihood of the crew's maintaining SA during the most critical stages of approach. However, I do acknowledge that my views necessarily are biased due to that routine exposure.