There are a lot of factors that govern plant propagation. Trust me, almost every gardener wishes we could just stick 'em in the ground and have multiple plant babies! Let us look at a few of the factors:
Some plants will simply not like propagation through gels, root powders and other agents. They might need a lab setting with carefully delivered hormones. They might only like propagation by seed. Don't worry, roses are not in this category.
The season, how the cuttings are obtained, moisture, location of your propagation attempts all play a role. This, definitely affects the roses.
The type of rose - climbing, mounding, hybrid, etc. is another factor as well. Some might need hard wood for propagation, some might do better with soft wood, etc.
Gel versus rooting powder versus other growth media matter as well. I have seen wet cow dung being used successfully by my mother in India for roses.
Now, coming to what happened, I do not follow how all the roots fell off, but that accident probably distressed the plant. So sticking the rest of the plant with just a little bit of the rooting powder may not have been enough.
Setting the plant in water, and choosing a few branches to save while the plant was still alive might have been one way to try and save the plant. Also, when a plant is injured, it is best to remove flowers, buds and excessive branches helps the plant focus more energy on healing by rooting.
Typically, for general propagation, you go to the mother plant, look for healthy branches, while they are still on the plant, and select a few, typically, soft wood (for vertical roses, those that still have green on them) and take a cutting, slicing at 45 degrees facing towards the ground. For certain varieties, you don't want cuttings from the hard wood branches, identified by hard, brown color, especially because that could also affect the health of the mother plant.
Identify the type of rose and follow general guidelines, but try to experiment if the guidelines don't yield results.
There are variations, but I have had success with even thin branches, though the wait will be long, over several weeks (note, even with the one your mother threw away, a month might not have been enough). When propagating, I also like to remove all leaves, and you absolutely have to remove any flowers or buds, because you want the cutting to focus its energy on growing roots, and not feeding the flowers.
Propagation, is really about trial and error. You should not give up after one try. You will also hear a lot of anecdotes about how different things work for different people, as well as how there really is only one sure fire way of doing propagation. The truth is in the middle. And sometimes, even after they put roots and shoots, plants can give up and die. It is part of the process.
Next time, ask your mother to try to get cuttings, find a place away from the sun, good soil or growth medium such as the powder (the powder doesn't work for me) or gel, or whatever is handy.
Sticking the cutting into the medium, covering the top with a ziploc bag or translucent bag (to allow filtered sunlight) and placing it in a warm enough and light accessible location, but still away from direct sun (direct sun is never good when propagating, caring for injured plants, etc.) is a good way to start. Try 2 - 3 cuttings at a time.
Based on your successes/failures, a few tries might be needed. It will eventually benefit both your mother and the neighbor, as they can have many roses to share.
Yes, legality can be an issue. However, for private, non-commercial use, in many cases, you will be safe. It only becomes a problem, if after six months of success, your mom sets up a store front, and an online site, as well as an Etsy account and starts selling propagated plants, left and right.
If you have doubts, unless it is a known public domain plant, if you bought it at a run-of-the-mill store like Lowe's, Home Depot etc., assume that hybridized plants and ornamentals (anything with a branded Nursery/breeder's name, for example, David Austen) have some sort of patent issued to them. Nowadays a lot of tags will also say that, but again, most restrictions only apply to commercial use.
Personal Anecdote: In the summer of 2014, I had a hardy rose bush in a pot, "die". Everything browned out, and I set the pot aside to reuse the pot and soil aside, but for months, never removed the plant itself from the plant. One wintry day in November, I looked at the plant and my rose bush had happily sprung to life (I have a personal blog post about this, if there is interest). And, I promptly re-potted the sucker into a larger pot, gave it tons of plant food, and let it winter over in Zone 9B. Now, the plant is as vibrant as ever, producing beautiful red roses.
So, just keep experimenting!