What you're looking for is called "automatic gain control" or "compressor".
The idea is to:
Measure the signal's "amplitude"
Apply attenuation or gain to keep loudness constant
I put quotes around "amplitude" because several things can be measured. If the goal of the system is to prevent saturation of amplifiers or magnetic tape then the variable of interest is peak amplitude, or close to that ; a peak detector can be used. In your application it would rather be loudness, which is more related to average power. As an approximation, average amplitude of absolute value can be used too, so that would be a precision rectifier followed by lowpass filtering.
To apply gain or attenuation to the signal in the analog domain, you need a voltage controlled volume control, which multiplies the audio signal with the gain control signal. This gets a bit hairy, because analog multiplication is difficult and expensive to with both precision and low distortion. Fortunately, DC precision is not required becaus there isn't any DC in the signal, and out of the box precision is not required either because the device can be adjusted with knobs to yield the desired effect.
So most systems will either use some form of voltage controlled resistor (JFET, LDR...) to tweak the volume, or an audio voltage controlled amplifier which is basically an analog multiplier with some constraints removed (not caring about DC, offset, precision, etc, means it can be a lot cheaper).
How these two subsystems interact with each other is when the can of worms is opened. Because the control variable ("loudness") needs to be filtered and smoothed, otherwise the circuit would simply compress every high value in the signal and let through every low value, resulting in huge distortion like clipping. So it needs to measure average loudness over a short (but not too short) length of time, and attenuate the signal according to that.
None of these systems can see into the future, they only react to the signal, so sudden transitions from loud to quiet and vice versa expose the system's reaction time. This is called "compressor pumping" and it is very recognizable, sometimes used as an effect. When it happens, every time the drummer whacks, the compressor turns the volume down for a few tenths of a second then back, and that modulates the rest of the audio.
Nowadays this is done with DSP because multiplication is not a problem for computers. So the simplest solution to your problem is to add a compressor plugin to your media player, there are plenty of those. Unlike analog, in the digital domain if you are playing back recorded audio, the compressor can cheat, look ahead into the "future" of the track, and begin attenuating just before it switches from quiet to loud (and vice versa) which reduces artifacts.
Is this doable within a reasonable definition of “doable”?
I'm not going to give any schematics because I've never built one of these and if you want one that sounds good, then you should ask someone who built one that sounds good.
Multiplying audio by any signal that is not constant is a nonlinear process that will add distortion, the question is how much and how audible. And since the way we hear and process distortion into "good sounding" and "bad sounding" is not well understood, a lot of experimentation and fine tuning is probably going to be required. So while the end result is not necessarily a huge complex circuit, it'll probably require a lot of tinkering.
Note that oldskool mixes like your Queen example are recorded at normal levels, ie the peaks are actually peaks. To avoid clipping during peaks the rest of the audio is at a much lower level, so the dynamic range is preserved. This is best for a calm environment, but unsuitable for listening in a car, for example, where ambient noise will cover the quiet passages. More recently, the loudness wars have resulted in many recordings which are compressed to death, there is no dynamic range, which renders the music unlistenable except in a car.