Agree that ash-throated is generally paler bird, but then so are the others if we're focusing on the head area. In some ways br-cr'd fits better. I've seen this species a fair few times and I've never seen one with the head area as dark blackish as here. Always brownish. Also, Sibley says bill should be 10% longer than great-crested but here it's noticeably short and stubby.
>>However, the bill...
Admit I don't get your point here. The linked photo isn't great in this respect. If we squint we can tell that the bill base is in fact pinkish but the ambient light makes this difficult to see. I'd be wary of making judgements based on the opus photo.
After all, not sure what we're left with. Problem for +/- all options is the extreme darkness of the head/cap. Bill is a problem for great-crested and I argue for brown-crested. Lack of strong tertial black vs white edge is a problem for great-crested (, as is generally pale-ish under colour). Dusky-capped should be out of range and in any case has a paler cap and a thinner bill. Ash-thr has the best bill fit but should be a paler bird. I suppose it's still my bet but I'd happily defer to an expert.