Why does the branch cut of the self-energy begin at $2m$?

First, remember that the spectral density is written for states with zero total momentum.

That branch cut comes not from bound states but simply from unbound two-particle states that form continuous spectrum (you can always take particles with arbitrary opposite momenta which will give you arbitrary total energy). In contrast two-particle bound states usually form discrete spectrum. That mean that they will contribute as poles below $(2m)^2$.