Electronic – Why can’t I swap a TL071 with a 741

I'm starting to wonder whether the 1k resistors are too small, as they're smaller than the 2.2k output impedance of the microphone.

Those are the output impedance of the microphone. If you look at the mic capsule's datasheet you'll see an equivalent circuit:

I don't know why manufacturers always show the FET as a triangle. This is how it's actually configured:

So this is really a common source amplifier:

The output impedance of a common source amplifier is just \$R_\text{D}\$, the drain resistor, so when the datasheet says "output impedance (Zout) 2.2 KΩ", they really mean "output impedance of our example circuit".

With \$R_\text{S}\ = 0\$, the voltage gain of the common source amplifier is proportional to \$R_\text{D}\$, since the FET acts like a current source, so the resulting voltage is determined by V = I(FET) * Rd.

What resistor should you choose? It depends. Generally you want high gain in the first stage so you can lower the gain of subsequent stages, which lowers noise. The distortion also decreases as gain increases. You can't increase \$R_\text{D}\$ forever, though, there's a point at which current is too low and distortion increases and gain drops suddenly. Also, if your microphone is expected to pick up high SPLs, you shouldn't increase the gain too much or it will clip.

I don't know how to optimize the gain based on the parameters in the datasheet, but I'd like to know. For mass production, the gm of the FETs will vary from unit to unit (and possibly the FET type will be changed from one capsule to the next even though they have the same part number), so optimizing for maximum gain for a specific FET is probably a bad idea.

I know analog guys don't like to hear this, but a small microcontroller could easily meet all of your requirements. It would be easy to modify its operational parameters even after the hardware is built, and it would have by far the lowest overall parts count.

A built-in ADC could digitize the audio (as well as the voltages from your control knobs), individual outputs could drive the LEDs, and a single PWM output could drive your motor (with suitable output buffers, of course).