Taken most literally, what you are proposing is complete analog simulation of the entire electronic circuit. Analog circuit analysis is not cheap. Generally, every point in the circuit has to be calculated against every other point, at a time resolution fine enough for the smallest relevant signalling details to be preserved. The problem scales non-linearly with the more components there are.
Spice is a simulator that is widely used for this kind of work, and a 20 component circuit handling audio frequencies will run many times slower than real-time on my fairly powerful modern desktop. Even a simple pocket calculator has thousands, if not tens of thousands, of components.
So some "shortcuts and tricks" must be done. for example, with fully digital circuits, you can simulate them with the assumption that they are, in fact, digital. This can be done at the transistor or logic level. It's still very computationally expensive, requiring every node to every node be computed. This is almost feasible today with simple machines. See the Visual6502 Project for an example where this has been done with the entire 6502 processor.
There is a transistor-level simulation of the TIA chip used in the Atari 2600, which is infamously hard to emulate. There is a complete transistor simulation of the Atari 2600, based on that and Visual6502. It is very slow. The reverse-engineered layout was, however, used to inform the current TIA emulation module in the Stella emulator. It uses a mixed level of abstraction, and runs quickly on modern machines.
Finally, there is the most common kind of emulation, behavioural, where the internal implementation does not match the real device. If completely accurate, this is potentially interchangeable with a full digital simulation. But for any complex component, making such an emulator fully accurate without fully reverse-engineering the electronics is often impossible. Something like a CPU emulator may not emulate esoteric behaviour that occurs due to an unusual circuit layout, for example.
Your general simulator would have to be general enough, and accurate enough, for all the components to work in whatever configuration you might put them in. That brings me to a point raised by @TrevorBoydSmith in the comments. We do not have such information for many designs. They are generally proprietary, and potentially copyrighted. The above transistor-level simulators of the components in the Atari 2600, for example, were created by literally breaking open chips, photographing them, and semi-automatically translating this into connection maps. It is demanding, complicated work that gets harder the more complex and miniaturized the components are. Only a few ICs have been mapped like this.
In practice, mixing these approaches is often what is done in emulators aiming for very high accuracy. A sound chip might be partially simulated as an analog circuit, for example. As one commenter notes, some emulators including the Commodore 64's SID chip take this approach for the SID chip. There is the DICE emulator for very early video games such as Pong. A mixed approach is necessary here, as these games rely on analog components, but could not be simulated as an entire analog circuit including every transistor. DICE runs just about fast enough to play some of the implemented games at full speed on modern computers.
There are various emulation suites, of which DICE is an example, which do take a broadly component-level approach, mixing levels of simulation abstraction as necessary. MAME/MESS takes a primarily behavioural approach, but does break the designs down into components for easy reuse. It can be almost trivial to implement a new platform with MAME, if all of the components already exist and are 100% accurate in their emulation. (They often are not.) These emulation suites usually only simulate just enough for the specific machines they target to be emulated accurately.
In conclusion, your general simulator would have to be very complex, incorporating both general-purpose analog and digital simulation. Most importantly, someone would have to actually translate the many proprietary logic or electrical circuit representations of the circuits we want into this simulator.
So, theoretically, it is possible. But it is a software engineering and data collection/hardware reverse engineering problem of truly vast scope. It is unlikely to run at real-time speed. And it might forever be too computationally expensive to do for anything with extensive analog components.