Why do spectrum analyzers use envelope detectors?
After mixing and passing through the IF filter what remains is largely one small part of the spectrum that your SA is scanning through. If there's a signal present in this part of the spectrum, it'll be a sinewave and if you envelope detect it you are, in effect, measuring the peak amplitude of that sine wave. That peak signal is \$\sqrt2\$ higher than the RMS value so it's perfectly usable.

EDITED SECTION

There will be errors in assuming that the signal detected is a single sinewave. For instance if you use an averaging filter after the envelope detector and looked at what it told you when a single sinewave is in that part of the spectrum it would produce a reading that is 3 dB higher than the RMS. 

If on the other hand you had three sinewaves at 999 kHz, 1000 kHz and 1001 kHz it would produce a level that is only 1 or 2 dB above the true RMS value and this does lead to a small error.