If the "starter" plants from the cuttings rot away at the stem, they could be damping off. Planting the baby plant in sterilized medium watering with filtered water with a dash of hydrogen peroxide or cooled chamomile tea and sprinkling the surface of the potting medium with cinnamon may help. There is a lot of information on this site and online about damping off.
Treating the baby plant with systemic rose fungicide may also help. I would try a more dilute dose at first to not burn the fragile roots.
I have also tried unsuccessfully to root rose cuttings. I will try again eventually. One of my roses, Baby Blanket, successfully roots itself by layering. You could try layering the rose on purpose. Lay a branch close to the ground against the ground, hold it down, put some good potting soil on it and hold it down with a small flat rock or something. Leave it for a long while. The branch in the ground may eventually grow roots. Once it starts sending up vertical growth, check for roots, cut the branch to the main plant and baby the new plant.
I have also read about people wrapping part of the cane with rooting hormone and peat moss and plastic wrap. the wrapped part can grow roots and be a new plant.
I hope you manage to both save and propagate your favorite roses.
I tried growing roses organically with mixed results. I try growing new, disease resistant roses naturally at first. If they start getting blackspot or other problems, I add them to the list of roses needing systemic fungicide and sprays. Some of my roses are fine with only neems spray for aphids.
