Hmm. You do not fly a Cat I to an MDA you fly to a DA or DH. Cat I is a precision approach, MDAs are for non-precision. Not sure where you got this idea from.
When you ask 'why put a DH in an MDA box', I think you might be referring to one of two things. It could be the MDH which has always been there for those operating on QFE. Or it could be the DA/H for a Continuous Descent Final Approach (CDFA has always been an option for NPAs but the difference here is how you treat the minima). That is my answer.
It used to be that precision approaches were all flown to DA/H, and non-precision approaches to MDA/H. Precision approaches were ILS mainly plus MLS and PAR. Non-precision was everything else. However, in this century the powers-that-be have really pushed the CDFA over the 'dive and drive', which has led to the application of the DA/H concept to the non-precision minima. Also, GPS has bridged the gap in 'precision'. All good of course but the jargon is now quite confusing.