I look at this list of monsters, most I am aware of, none I have used, most I would not use (at least, not as presented), and I cannot help but think that there are more pressing questions than what creature type to best shoehorn them into.
For starters, do they really need laundry lists of spells, spell-like abilities, and maybe even psionics? Many of the monster entries mention casting spells like a cleric and/or wizard, with the phoenix getting called out as having "tons of magical abilities". Given that the coautl could cast spells like a 9th-level sorcerer in 3rd Edition, I wonder what the barometer is?
The mythological baku started out as a chimeric creature that could protect against disease and evil. Later dream-devouring would get added to its roster, and its appearance would eventually change into that of a tapir. The concept and capabilities seem fairly simple, so what is with all the the psionic powers? I no longer have a copy of The Complete Psionicist's Handbook, so I can only guess that absorb disease makes sense, but what about animal affinity, metamorphosis, and ectoplasmic form?
For this I greatly prefer 4th Edition for monster presentation: you get an entire block, on the same page, and you do not have to reference one or more other books to figure out what anything does. Just a glance at the "typical lammasu's" loadout of twenty-one cleric spells (of which a merciful few were prepared twice) causes flashes of Edition-induced PTSD.
When it comes to type I disagree that Dungeons & Dragons, at least in both 3rd Edition and Next, have an orderly structure of creature types. For example, in 3rd Edition if you came from another plane, you were an Outsider, but inevitables are considered Constructs. Even mortal, humanoid races like aasimar and tieflings were Outsiders, unless they were born and raised on the Material Plane, in which case they were Outsiders with the Native subtype.
Next not only retains some artifacts from 3rd Edition, but makes things even more confusing. Why does Giant have to be a separate type from Humanoid? I always figured that a size category, or maybe even a subtype would do the trick. Do Fiends need their own type, when there are Demon and Devil subtypes? The same thing goes for the Dragon type: it seems like Monstrosity would do, despite it being divided from Beast by a very, very blurry line.
While some of the monsters in the article clearly come across as celestial entities, like the hollyphant and ki-rin, that does not work for all of them. I think that the baku and couatl would work just fine as magical beasts, though this might be grounds for a spirit type, or at least a subtype. Again, 4th Edition did it better with its origin/type combinations, and with only ten options between them both to boot. You could give them the immortal origin, lock them in a tomb, and call it good.
As for the poll questions, while I do not think that they necessarily need to get rid of any of them, I think a lot of them could stand to have their concepts and/or mechanics revised. For example the lamassu and shedu could be folded into one monster (since they are both the same thing), and you could stick with the bull depiction to help differentiate them from a sphinx.