I don’t yet understand why 2 energy levels aren’t enough as they can still emit photons (even when B12 and B21 are equal). By the way, I’m talking about stimulated emission.
In a two-level system you can have stimulated emission, but you can't have gain. No gain, no laser amplification, no laser.
Stimulated transitions work both ways with equal probability: up and down. As long as there is more population in the lower state there will be more "ups" than "downs", that is more absorption than gain. The best you can do is to introduce a very high intensity "pump" beam which will induce as many "ups" as possible. You can get the populations of the two states to be (nearly) equal, but in that case the rate of "ups" will equal the rate of "downs" ... again no gain.
To get gain you need an alternate way to populate the upper state: a third level.
Albert Einstein showed that rate of stimulated emissions is equal to that of stimulated absorptions. That is, a photon of energy $h \nu$ has equal probablity to cause either absorption or stimulated emission. Suppose there are only two levels;one ground state and one metastable state. Then proportion of stimulated emissions will be equal to that of absorptions. Therefore no light amplification will occur. Instead if we are exciting atom to a higher energy state above metastable state, the atoms at higher state will eventually fall to metastable state causing population inversion. In this case, number of stimulated emissions will be greater that of absorptions. Then we will have light amplification