I think it's a great question! And I like your 2-to-3 word descriptor idea.
I've been playing around with a totally redesigned stat block, but one thing I've been looking at is what gets priority? What do you see right up top?
One thing I think deserves a place clearly up front in the stat block, before even essential things like HP, AC, speed, and so forth – is the monster's Motivation. What does it want?
This doesn't have to be a singular thing, though it could be for a particularly simple or focused monster.
For example, for a giant spider this might be:
• Ensnare easy prey to devour later.
• Protect its eggs, especially from fire.
The reason I'd like to see motivation right up front is that it suggests ways the PCs might interact with the monster either outside or combat or leading-into-combat. The PCs might threaten the giant spider egg cache with fire, causing the spiders to delay attacking. Or the PCs might create a decoy of prey with a conjured beast or an illusion, so they can sneak past the spiders.
For something like a giant spider, the motivations might not inspire many adventure ideas – ok, yeah, it's got cocoons with trapped victims and there's probably an egg cache somewhere. But it starts getting more interesting the more sentient the monster is.
For example, a manticore's motivations might be (d6):
• Remove the tail spike from its paw.
• Find its missing pack.
• Chase off or kill a territorial rival (chimeras, griffons, perytons, wyverns).
• Receive tribute in the form of delicious meats or ale.
• Avoid dragons at all costs.
• Serve its master well.
I thought the Motivations by Creature Type in Tasha's were a big step in the right direction, but really this is such an essential piece of a monster that it should be presented first thing in its stat block.
Motivation does what alignment & personality/ideals/bonds do not – it's where all that higher concept stuff gets translated into something the DM can use immediately at their table.