Estella - I'm sorry you had this bad experience with an otherwise lovely plant. Buying plants in Chinatown is always a crap-shoot as many of the shops there don't have experience with plants and are only concerned with generating quick sales. They should have made good on your plant especially since you purchased it so recently.
From your description, it seems that the stem or trunk cutting was just recently taken and stuck in soil and sold long before any roots had a chance to develop. Or else, it did have a good root system, but the shop owner failed to properly care for it so the roots were already dead when they sold it to you. Well rooted plants will, not fall out of their pots.
Your Aralia is not sick or diseased; it simply does not have any roots to nourish the plant. There is no reason to quarantine it.
Trying to get it to develop a good root system will not be easy and will take a long time. You will have to keep the soil barely damp at all times by watering lightly as soon as the top inch of soil is dry. Place it in a warm, very sunny location close to a window. The problem is that it will continue to lose leaves and stems until new roots are fully formed and that will take many months.
An alternative is to take some smaller stem cuttings from the remaining healthy stems and root them in water.
If you can gently unpot the plant and take a clear photo of whatever roots are at the base of the stump, I might have some other suggestions.
If the retailer in Chinatown has a presence, consider writing a bad review on Yelp so others will learn from your experience.
For the future, you might consider plant purchases in the flower and plant market on 28th Street.
