Hey! Scientist in training here (Biomedical).
I think you pose an interesting question, and I decided to do some research, and share my findings. I've just finished watching the documentary before commenting, I felt that was the least I could do.
To begin with; I don't believe the classic flying, fire-breathing fantasy dragon ever existed. There are no records of the required sorts of adaptations in any related species in the fossil record or in current (descendant) species. I'll explain what I mean by taking the bombardier beetle mentioned in the documentary as an example:
beetle
The bombardier beetle launches two chemicals, that when they connect react to reach a boiling temperature. There are various kinds of beetle that have the precursors necessary for this chemical mixture in their body, but do not use them as a weapon; one of the chemicals is present in nearly all beetles as a hardening agent used in their exoskeletons, and the other is a foul-smelling substance that is a deterrent to being eaten. Bombardier beetles are special in that they eject these two substances to a weaponized effect, but the chemicals themselves are actually quite common, and even sometimes stored inside sacs like the bombardier beetle does (but not ejected and mixed).
As such, the bombardier beetle is, in my eyes, actually a very strong argument as to why dragons couldn't have existed as they are imagined, rather then a supportive argument to the idea.
platinum
I also have a problem with the platinum / hydrogen / oxygen mixture, namely that platinum is extremely rare. It has an average abundance of 5 micrograms per kilo; this comes down to 5 parts per 1000000000. When platinum is found in nature, it is usually with nickel, copper, or heavier metals - that is to say, it's poisonous, and consumption would likely lead to metal poisoning. There are some areas where rivers have washed out platinum from the rock, e.g. in Africa and Alaska, but I can't see how this would ever support an entire species, or prompt a species to evolve a dependency on such a rare metal. Especially once we see the dragons in the ice age, where there would be no flowing rivers in those frozen areas, I don't see them getting to any platinum in any way.
hydrogen
The proposed hydrogen-filled sacs inside the dragon would not matter to any great extent. The lifting capacity of hydrogen is far too low to make any significant difference to such a large, heavy animal - it would not even make them walk more softly, as the forest dragon was shown to do. Assuming the dragon, taking all the hydrogen it can muster - in the bones and various sacks - comes to having 5 cubic meters of hydrogen in its body (a very generous estimation). The lifting force of hydrogen is 1.114 kg/m3 - so it would give it a 'lift' of roughly 5 kilograms. On a body of 600 kilograms, that's slightly less than 1%. Using hydrogen for aerial lift is not a winning proposition.
large fossils
Optimistically; Megalania may have been contemporary with the first Australian settlers, 50 000 years ago. It is the largest lizard that has been found in the fossil record so far, with an estimated length up to 7 meters and a weight of up to 600 kilos. This is still nowhere near the 12+ meters of the tyrannosaurus rex, at an estimated 11 000 kilos, but it's a large lizard (possibly) contemporary with humans.
If we ignore the overlap with humanity, the largest flying creature we know of was the Quetzalcoatlus. With an estimated wingspan of 16 meters (52 feet) and recently a larger bodyweight of 200 kg (rather than the initial estimates as low as 70 kg), it proves such large flying creatures are at least in theory possible. However, large flyers evolve to become as light as possible - meaning an armoured flier is again very unlikely.
conclusion
So biologically and evolutionarily speaking, dragons combine traits of animals that ended up taking very different, almost opposed developmental paths. For these traits to come together naturally in a single animal is unlikely, very inefficient, and unheard of.
Theoretically speaking, though, it should be possible, assuming a completely arbitrary level of genetics expertise, to create dragons. It could very possibly prove to be a completely uncompetitive creature, possibly unable to survive in the wild, but theoretically if one were to give it hollow bones, arbitrarily light scales (something including spider-silk proteins, perhaps), maybe the respiratory system of birds (which is fascinating, and completely worth looking up on its own merit) and humongous wings, and heck, add your favourite set of mammalian curving horns - there is no -hard- reason why a dragon could not be created. There is no single aspect of the creature that makes it impossible (as an added bonus, it would not only be a dragon: since it has DNA from various creatures it would also be a chimera!).
Please do note that the scientific community is nowhere near doing such a thing, perhaps they never will be able to, and if the level of bioengineering is ever achieved it is very likely that no one will never bother to. Do not hold your breath on ever seeing a physical dragon in your lifetime.
ps
Dropping the scientific angle completely, I do have a suggestion as to why so many cultures on the planet have depicted dragons in their art and mythology. For those who spend time in realms other than our daily physical world, encountering dragons, whilst by no means common and often unsettling, is not unheard of. Dragons exist and have likely existed in the "higher" realms since at least the beginnings of recorded history. Everyone has access to these places, many traditions believe dreaming to be the most natural and uncontrolled way of "transcending" the body. Via magic it is possible to consciously connect to these realms and their non-physical inhabitants at will, and communicate with them. The most comprehensive system of ritual magic I know of is the Golden Dawn system. Although I do not practice that system myself, it might be interesting to research. I am sure there are myriad other systems out there that allow for the same results. There are Christian systems of magic, and (at least) one for most other religions as well as entirely non-religious systems. Chaos magic in particular uses lore and explanations that tend to be more satisfactory to the scientifically minded. I do not practice chaos magic either, and can't personally advocate for any of these systems. Learning any effective magic system will take several years, but it may well take you to the origin of the dragon mythos, and explain many other things about the world along the way.