At the epoch of (re)combination, the radiation field of the universe was a blackbody at a temperature of roughly 3000 K.
I am not sure where you get the idea that the temperature is directly given by the binding energy of a hydrogen atom - that is not the case. A blackbody at 3000 K has an average photon energy of 0.7 eV, but any hotter and there are enough photons with energies above 13.6 eV to ionise any H atoms.
Prior to recombination, the universe consisted of free electrons, protons, helium atoms and a very small number of hydrogen atoms and alpha particles in thermodynamic equilibrium.
Since the plasma is in thermodynamic equilibrium all emission processes are balanced by absorption processes and the source function is given by the Planck blackbody function. In addition, since the mean free path of a photon is much smaller than the speed of light multiplied by the age of the universe, mainly due to the opacity provide by free electrons, the universe is effectively optically thick. In such circumstances, the radiative transfer equation tells us that the radiation field approximates to the source function, which in this case is the Planck blackbody function.
The recombination occurs at 3000 K because at that temperature and below, there are insufficient photons with high enough energy to ionise lots of hydrogen. You can calculate the ionisation fraction using the Saha equation - this shows that the ionisation fraction decreases abruptly at $\sim 3000$ K (see here for some details). As the hydrogen atoms recombine, the opacity falls by orders of magnitude due to the disappearance of free electrons and the radiation field that exists at that time is then free to propagate throughout the universe.
Your real question might be why can a gas consisting of electrons protons and atoms produce a continuous spectrum? The answer to this is that there is enough opacity at all wavelengths to make the universe optically thick at those wavelengths and that absorption and emission processes must be in detailed balance. The relevant processes that absorb photons over a continuum of wavelengths are electron scattering, inverse bremsstrahlung, and photoelectric absorption, with inverse continuum emission processes of inverse Compton scattering, bremsstrahlung and recombination radiation.