Snakes and spiders can’t be trained as much as dogs to keep to themselves and behave appropriately in crowded places.

A behaved dog doesn't rid someone of the phobia, unfortunately.

Snakes and arachnids can be incredibly docile, more so than a trained dog. But would it be fair for someone to bring a snake around, knowing that the fear of snakes would affect a ton of people who simply want to go on a flight? Seems odd to create new problems in this situation.

If they kept these dogs out of sight, but with way finding signs to get to them, should someone choose to, then it's less of an issue. You'd still have to wonder whether introducing an allergen to a public space is a good idea or not.