Trapped-ion and neutral-atom QC require lasers because the light signal needs to be coherent. That's the main feature of a classical laser, really. The explanation with the number of photons doesn't really cut it, because even a perfect laser does not have a definite photon number: coherent states are inherently uncertain in both photon number and phase. But LEDs are even worse, because the light signal is truly incoherent. It's not even a good quantum state, it's a classical superposition of incoherent photons that you can't really use for any quantum control.
But even more than that, this seems to me like a purely on-chip solution. For trapped ions and neutral atoms you really need to translate to free-space optics at some point.
Indeed, it is nuanced, as you point out. For example, you can't just attenuate a laser and use that as a single photon source (instead you'd get a coherent state). To realise a true single photon source you need an additional (quantum) process, like controlled stimulated emission from single atoms, or driving some nonlinear crystal to generate photon pairs (that's using spontaneous parametric down conversion, i think). And that's where the coherence properties of the laser are essential.
As for fully integrated optics, it's where quantum computers eventually want to be, and there's no physical limitations currently. But perhaps it's too early to say whether we would absolutely require free space optics because it's impossible to do some optics thing another way.
But even more than that, this seems to me like a purely on-chip solution. For trapped ions and neutral atoms you really need to translate to free-space optics at some point.