How can I make my hard stone entrance staircase softer and more kid friendly?
Carpet the stairs. A fairly thick outdoor carpet could be secured using high-strength construction adhesive along with screws into the metal risers under the lip of each step. Perhaps even a rubber "nose" under the carpet, along the sharp edge of the steps. That would lessen the danger of a fall, at least a bit.

When unbearably muddy/dirty it could be water blasted and wet-vac cleaned.

When the tykes have grown the carpet could be removed (albeit with some labor to scrape off the glue and patch work on the screw holes) to reveal the nice original stone steps.

Contact the local flooring industry council or guild for advice; they would love to offer solutions and supply and install it for you. Or you could take a chance and do it yourself.