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.
Down syndrome is an example of that anomaly rather than some mutation that’s passed down. It’s an extra partial or complete copy of chromosome 21, and therefore can happen to any couple.
Think r/K evolutionary strategies [https://en.m.wikipedia.org/wiki/R/K_selection_theory]
A species that can produce 1000+ offspring (or more) per mating (with multiple matings per adult lifespan) - like most insects, seaturtles/squids, etc. has little incentive to have high precision/accuracy reproduction.
One that does a small handful (whales, elephants, etc), has a lot of incentive to do so (or self abort early on if there is an issue).
This gets complicated though, because while a single individual in a species which is predominantly r strategy (few offspring), can be ‘more’ on the K side, and vice versa.
And a species which has too much consistency, both genetically and ‘approach’ is very susceptible to inbreeding/mono-cropping issues where a single event/disease can wipe them out, or they can even destroy themselves due to recessive traits.
So there is a general (but diffuse!) evolutionary pressure towards a degree of mutation/error in reproduction in even the most hardcore ‘r’ species (which will necessarily produce a lot of noise/‘wasted paths’) at the species level, alongside a hard evolutionary pressure to reduce it for individuals.
In for example humans. Or elephants. Or whales.
This is also why things like rich/poor, healthy/sick, pretty/ugly, strong/weak, etc. will never go away - they’re outcome distributions along a probability curve due to fundamental different approaches by individual humans due to the necessity of how humans have to be in order for humanity (the species) to survive.
Individuals have strong incentives to ‘tip the scales’ in various ways of course, and societies (at a minimum!) have strong incentives to stop them. It’s why aborting based on gender is illegal in India, for instance. [https://pubmed.ncbi.nlm.nih.gov/7920120/]
Because if everyone had the same attributes in order for everyone to be ‘pretty’ or ‘strong’ or ‘rich’ or whatever, then some hypothetical weight-lifting-infectious-disease (or a famine, or an attack by jealous anti-weightlifters) would wipe out the whole population. And if everyone optimized for ‘rich’ genes (whatever that means), then society would implode, because that literally couldn’t work.
And those types of situations, albeit less light hearted, have, do, and will continue to happen eventually.
Turner syndrome is another example without the mental issues: https://en.wikipedia.org/wiki/Turner_syndrome
In the larger context, there may be some long term advantages to chromosomal anomalies because their number varies so much between closely related species.
As far as I understand it, we now have the technological capacity to do so, and simply lack the moral and legal capability.
For me this would push occams razor toward the notion that endeavoring to build monumental generational-construction-project tombs was not an act of cultural vanity but actually an informed and practical decision made with some form of ancient knowledge.