For those who feel weird about the whole "forbidden transitions being only possible with quantum tunneling" thing and want an alternative interpretation:

It's only true that the transitions are forbidden under a given simplified model of the atom. It is very much possible to calculate the transition probabilities under a more realistic model, and the previously "forbidden" transitions are now just regular transitions that occur with lower probability.

In this case, the simplified model is that of the electric dipole approximation, where the atom is taken to be an electric dipole (reasonable when the wavelength of light emitted during an atomic transition is much larger than the size of the atom).This means it interacts with electromagnetic radiation only through electric dipole interactions, which implies that energy transitions must change orbital angular momentum, hence the 21cm transition is "forbidden". However, in reality, the atom is not truly an electric dipole, and so the 21cm transition is possible by the magnetic dipole interaction, just with low probability. (This low probability is due to the relative strength of the magnetic interaction compared to the electric interaction).

I've never liked the definition of forbidden transitions as "transitions not predicted under the broader approximation", because its rare that anybody actually lays out why a given approximation is used, and therefore why that approximation is inappropriate for the "forbidden" situation.

The reality is that with e.g. 21 cm Hydrogen, or 500.7 nm Oxygen (which I knew by heart, back in the day), its hard to keep a given atom in the appropriate state long enough for it to relax by emitting the appropriate photon. Indeed, we can't create a pure enough vacuum in a large enough chamber that such things happen frequently enough to be measurable.