OK
so if I understand the contributions to this thread, building an HST that could run off OHLE isn't possible because of the complexity of a transformer to convert the power to DC for the traction motors.
So.... would it be possible to have an electro diesel hst that could pick up from the 3rd rail?
The transformer doesn't convert from AC to DC - that would be a rectifier (of which there are many different types). A typical modern scheme would be:
1) 25kV/50Hz AC Single phase supply
2) Transformer to step down to something in the ~1000 to 1500v range (still AC)
3) This is then fed to a rectifier (or pairs of rectifiers), thyristors, GTOs, IGBTs, etc.... The thyristors (or similar) will operate a controlled "firing angle" (google that if you want to know more - here is not the place) controlled by a computer which results in a DC link voltage - sometimes fixed, sometimes variable.
4) This link voltage is then used in many ways. Some is used via a chopper or similar to provide "hotel power" or "ETH" depending on your preferred nomenclature (basically, lights, heating, A/C, etc... for the carriages), and of course the traction systems.
5) If the train has 3 phase AC motors, the traction system will take this DC link voltage and via an inverter and variable voltage and variable frequency control (VVVF), provide what the motors need. There will also be a supply for things like motor blowers and if there is some kind of dynamic braking, it usually comes from/goes through here too.
So the transformer is only the first stage of things (or second, after the pantograph and main breaker). It is also a comparatively simple device compared to what happens later in the system, although the transformer usually is the single heaviest component on the train and requires significant cooling which usually adds an oil circuit, radiators, pumps and fans into the mix too. All of that adds weight and takes up space. This is your main problem in this fanciful HST conversion that will never happen.
The rectifiers, choppers, inverters, rheostats and control systems that are needed later in the description above all have very specific requirements in terms of the space they can be fitted into and their own cooling requirements.
In answer to your final point - there has already been a 3rd rail electro diesel - class 73 I think (someone may correct me). It was pretty gutless off the 3rd rail, but it did have that capability, and if there are any still running anywhere they must be about 40+ years old. Again, google is your friend here.