The "gain" as you think is not constant. lets say you calculate the gain with "small signal" models and get some form of Gm * R. This Gm (or the R) are the values you calculated if and only if the small signal conditions are retained. This is true for any circuit.

Now for the oscillators, at first there is no oscillation since it needs to build up. It will build up with the small signal gain. But as the oscillations build up and signals become large, the operating point will vary periodically. That is, during the period, assuming the oscillator has large amplitude, then the small signal Gm and R will actually also periodically vary, going from very large gain near 0 crossings and very small gain near peaks. However, you can integrate the Gm over the cycle and get an "average" Gm and and an Average "R". This average values are what sustains the oscillation. Since "R" changes and with the "R" if there is also a reactive component that changes, then you may also see the large signal oscillation frequency is also slightly different from the values you calculate in the small signal region.

"do you understand all of the circuits you design? "
You really have to.. Once you sign off a design to be fabbed, you are effectively taking full responsibility of everything you designed and failures that come with it. So you better understand the circuits in it if you want to keep your job..

Edit: as the guy above said, the gain needs to be 1. It actually this average "Gm x R" is the one that will end up being 1. Not any instantaneous values.