Would it be possible that only one of the three chromosomes is passed on from each parent, leading to a normal and healthy child?
The primary cause of Down syndrome is trisomy of chromosome 21, which is what you described. That form is not inherited. A much less common cause, mosaicism, is a random occurrence during cell division and isn't inherited either.
The only form that can be inherited (that we know of) is caused by translocation. Basically a piece of one chromosome can become switched with a piece of another chromosome. If no information is lost, that's a balanced translocation. Those usually still produce a normal phenotype. When the balanced translocation is passed down a generation, it can lose information or pick up extra information and become unbalanced. If the extra information is from chromosome 21, that can become Down syndrome. That's a simplistic explanation, but should do the job.
A similar phenomenon is common in plants. While humans have two copies of DNA, one from each parent, many plants contribute more than one copy of DNA to their offspring, resulting in plants with many more than just two copies of DNA. The number of copies a plant has can vary within the same species. Notably, domesticated plants often have more copies of their DNA than their wild counterparts have, yet they are still reproductively compatible.
In animals like us, multiple copies of DNA, or even just of individual chromosomes, are much more risky, which is why it's so rare (those embryos are typically nonviable).
However, different animal species do often have different numbers of chromosomes, which does provide a pathway to evolutionary divergence and speciation. Typically, species with different numbers cannot reproduce (e.g., humans and chimps), and when they can, the result is often sterile (e.g., horses and donkeys).
There are several ways that species can gain or lose new chromosomes as they evolve, and you are right: duplication of an existing chromosome is one of them, although pretty rare because it often leads to serious health problems or sterility. (More common is the splitting of a chromosome or the fusing of two chromosomes, since these preserve all the genes without duplication. Another possibility is the total deletion of a chromosome, but this is extremely rare because it typically kills the embryo.)
So yes, theoretically, fertile men with trisomy 21 Down's syndrome (they are usually sterile, but there have been exceptions) and women with trisomy 21 Down's syndrome could get together and start a new hominid branch. Over time, the twin chromosomes would undergo selection and diverge from one another, likely in a way that would improve on the general health and fertility issues that people with this syndrome commonly struggle with.
The chances of this happening are very low, but it is certainly not unprecedented in the history of evolution.
Of course this niche doesn’t exist everywhere, for example in Iceland they have a very aggressive abortion campaign against the syndrome.
Iceland has extensive support for disabled people though, irrespective of abortion policies. These are of course not mutually exclusive things.
Besides there is already a social taboo against engaging in sexual intercourse with them because of the substantial power asymmetry. So really the only impediment to speciation is the lack of fertility.
The bad news is that the prognosis is even worse than for the trisomy.