I think you are worrying unnecessarily. Commercial orchardists routinely replace dead apple trees with a new one in the same spot. Very few diseases of any significance to apples will remain present in the soil and most of those can be circumvented by selecting a disease resistant cultivar (which Honeycrisp is) and by the appropriate rootstock. Your biggest worry is going to make sure you have the stump and roots of the old trees out of the way to make planting easy.

btw, mummified fruit can just be as simple as fruit that never fell or eaten by any creatures and has started to deteriorate/decompose on the tree. And most old, untended apples will look 'diseased'......it is just a symptom of old age :-)

Yeah...forget about and plant.