Simplest "physical" photon generating Feynman diagram

I’m half way through the excellent “Student Friendly Quantum Field Theory” and I read that single vertex Feynman diagrams in QED are “not physical” because their corresponding amplitudes are zero. For example the diagram
$$
e_{mathbf{p_1}}^- + e_{mathbf{p_2}}^+ to gamma_{mathbf{k_1}}
$$
has a probability amplitude that (when you calculate it) includes a factor of $delta^{(4)}(k_1 – p_1 – p_2)$, where the boldface p’s and k are 3-momenta and the normal typeface are 4-momenta. The argument goes that since the photon is massless we must have $k_{1mu}k_1^{mu} = 0$, but if you work out $(p_1+p_2)_{mu}(p_1 + p_2)^{mu}$ it turns out non-zero, therefore we can’t find a real photon momentum that makes the dirac delta, and consequently the amplitude $langlegamma_{mathbf{k_1}}lvert e_{mathbf{p_1}}^- e_{mathbf{p_2}}^+rangle$ non zero. A similar reasoning shows that every other single vertex Feynman diagram (e.g. $e_{mathbf{p_1}}^-to gamma_{mathbf{k_1}} + e_{mathbf{p_2}}^-$) are also “non physical”.

So my questions are:

1. If these diagrams are non-physical what’s the simplest diagram that generates a photon that is physical. Or, “where do all the photons come from?”
2. Are there any interpretations (for the fact the amplitude for single vertex diagrams are zero) other than that they are “non-physical”? For example, perhaps photons with $k^2 ne 0$ are possible, but live too short a time to be observed.

Please be gentle, I’m not actually a student, just an enthusiast and this is my lockdown reading!