How do you calculate the required gain for an electret microphone amplifier?
Your math is about right but what's missing is that 

(a) 1Vrms is 2.8Vp-p for a pure sinewave and well over 5Vp-p for normal speech, so (even without headroom for speaking a little louder) 1Vrms is too much output from your mic amp into a 5V range ADC, you probably want some headroom - maybe6-12dB, so 0.5 or 0.25V rms for a 60dBa input.

(b) if you're measuring normal speech as 60dBspl, you're measuring it from some distance away - like, across a room. Or at least 1 metre.

So if you're building a bug for espionage purposes, a gain like 1000-4000 is probably what you need.

But for other purposes, speak into the mic and you'll need much lower gain, like the numbers you're seeing from other sources.

(c) Sound level control is HARD - for top sound quality you need an experienced guy continually monitoring and adjusting the gain. for other purposes, don't imagine the problem stops with the mic amp - often it's just the first gain stage (and optimised for good noise performance).

Later stages involve more gain, and frequently some way of adjusting the gain, like a compressor, limiter to keep the audio levels within acceptable limits.