Is it possible to build a very minimalist radio transmitter that can be built only with passive components, what would be the schematics for that?
Sure. You can switch transients into a resonant circuit, as others have mentioned. The trouble here is that you end also transmitting a lot of those transients, which means a lot of broad-band noise. There's also the trouble that spark-gap transmitters are not permitted by international treaty. (Not that regulations matter much in a hypothetical post-catastrophic scenario, unless it's the authoritarian regime with a special emphasis on radio regulation variety)
Interestingly, one can make a CW transmitter (that is, one that transmits just one frequency, without the broadband noise) with passive components. All you need is a high frequency voltage source, which can be generated electromechanically.
For a real, working example, see the Varberg radio station. It is, quite simply, an AC motor, coupled to a generator to a 1:3 speed-increasing transmission, with the generator having 976 poles. The generator spins at about 2133 RPM (35.55 revolutions per second), and with 976 poles that makes at output at \$ 35.55 \cdot 976 / 2 = 17.35\:\mathrm{kHz} \$. The generator's output is switched into an antenna to transmit a tone, or into a short for the space between the dits and dahs. There's a bit of additional stuff to provide a matching network and control the motor, but it's all passive.
As for the premise of the question, that passives are easier to scrap, that may be somewhat true. However, discrete active components, like transistors, are not so hard to scrap either. Certainly, easier than building an electromechanical transmitter. A transistor radio can run off batteries, is more portable, and energy efficient. So in most ways that I imagine a post-catastrophic world, I would probably end up with a transistor radio.
Although the FETs in on a monolithic chip are symmetric, many discrete FETs have a very different structure which tries to maximize the usable surface area as well as source/drain connectivity. The bulk substrate connection on a transistor or chip has excellent current-handling capability, and if one were designing an NMOS LSI chip in which every single transistor needed to have its source or drain tied to a common point, performance would probably be optimized by having the substrate serve as the source or drain for all the transistors. Most chips, however, use the bulk connection as a common base, wasting its current-handling abilities, but allowing the source and drain connections of each transistor to be independent.
A typical "discrete" MOSFET will in fact be not one transistor, but dozens or hundreds of transistors in parallel. Because all the transistors are supposed to have their drains tied together, using the substrate as the drain won't cause the same design problems as it would in an LSI chip. Since the substrate can be very well solidly connected to an outside terminal, such a design will both improve drain conductivity, and also eliminate the need to use top-side metal for the drain connection, thus allowing the use of more metal to connect the sources. Unfortunately, if the transistors are arranged so that all their sources form a "mesh" (good for connectivity), that will leave their bases as isolated islands. While it would be possible to run metal tracks to connect all the bases together, doing so would require either subdividing the source-connected metal into many strips (degrading performance) or adding an extra metal layer and an extra insulating layer (significantly increasing cost). Since each base section has the metal layer for the source connection sitting directly above it, it's much easier to simply have have the bases as well as the sources connect to that.
Best Answer
There are four physical quantities of interest for electronics: voltage, flux, charge, and current. If you have four things and want to pick two, order not mattering, there are 4C2 = 6 ways to do that. Two of the physical quantities are defined in terms of the other two. (Current is change in charge over time. Voltage is change in flux over time.) That leaves four possible relationships: resistance, inductance, capacitance, and memristance.
If you want another fundamental component, you need another physical quantity to relate to these four. And while there are many physical quantities one might measure, none seem so tightly coupled as these. I'd suppose this is because electricity and magnetism are two aspects of the same force. I'd further suppose that since electromagnetism is now understood to be part of the electroweak force, one might be able to posit some relationships between the weak nuclear interaction and our four elements of voltage, current, charge, and flux.
I haven't the first clue how this would be physically manifested, especially given the relative weakness of the weak nuclear force at anything short of intranuclear distances. Perhaps in the presence of strong magnetic or electrical fields affecting the rates of radioactive decay? Or in precipitating or preventing nuclear fusion? I'd yet further suppose (I'm on a roll) that the field strengths required would be phenomenal, which is why they're not practical for everyday engineering.
But that's a lot of supposition. I am a mere engineer, and unqualified to speculate on such things.