This turned out to be more of a general limiter-info post rather than an answer to your question, but hopefully someone will get something out of it.
Imagine an “ideal” limiter, a function that leaves no samples above your target threshold, the simplest type would have instantaneous time constants (attack and release). This would equal hard digital clipping, pushing your master into the red. The more you push the signal into the red the more your waveforms change from sinusoid to square-like. Square waves consist of odd harmonics, the first one being the fundamental harmonic, the second one being the third harmonic, then the fifth, seventh and so on. This type of signal sounds very harsh, the ear does not like distortion with a loud third harmonic component.
To overcome this problem, a limiter made for audio signals introduces time constants, most notably the release time since in the typical way a limiter is used, it would not be of much use with anything else than an almost instantaneous attack.
Now you have to make tradeoffs between release time and effectiveness. Too short a release and you’ll more or less hard clip the signal, too long and you’ll lower the total volume of your track, just like lowering the master fader. Not to mention all the pumping in between.
When using a distortion effect you are looking at the whole thing from a different direction, the reduction in dynamics is a byproduct of the waveshaping taking place and it is not very versatile as a limiter. In comparison with the first example, a distortion effect works in quite the same way regarding time. The difference being that it typically introduces a lot of second harmonics, sounding subjectively good to the ear, but also components higher up the spectrum which will sooner or later sound harsh too. In comparison with a normal audio limiter, you will, as you almost say, have to introduce a ridiculous amount of distortion to achieve an equal amount of limiting, and I’d not want to listen very long to such a mix.
A subtle amount of distortion in combination with a carefully adjusted limiter might be better practice (a lot of people do this by running their mixes onto tape to get saturation distortion or through tube amplified -insert random gizmo here-), but I always recommend people to limit their mixes as little as possible, dynamics are awesome!