But from my perspective on the ground, there wouldn't be much point in you commencing an approach if the cloud base was below the DH minimum for the category of approach that you are flying
Cloud base only affects non-precision approaches. An ILS approach and the rules surrounding the legality of making any such approach are governed by RVR. DH affects CAT II/III operations so cloud base is even more irrelevant in this case. As mentioned low viz ops require a certain visual reference, in CAT II/III ops this is going to vary between runway lighting/elements of the approach lighting down to one runway light or even no visual reference required for CAT IIIB. Cloud base has no bearing in these circumstances.