Now for the big long answer: I don't like the three pillar model at all. After all, it's not a revealed truth of gaming in general, it's a design philosophy for 5e DnD.
I think there should be two pillars: Dungeons and Dragons
Dungeons is the interesting, exciting, mysterious, and treasure-filled locations where adventures can happen: yes, caves and castles, but also forests and deserts and mountains and elemental planes and fey wilderness and the pits of Hell - and cities large and small and small out-of-the-way villages and caravans and circuses and really anywhere w=something that could be called an 'adventure' could take place. By thinking of settings this way, you can focus your mind on making them good for adventure - which means more than traps and tunnels, but also means there are absolutely traps and tunnels and all the other fun things to find.
Not that there shouldn't be safe places to rest - but these only serve the game through the contrast they provide.
Dragons are the weird, dangerous and occasionally evil beings you find along the way - but most importantly, they're not there to fight and there not there to talk to: both of those things (and sneaking around and running away) are clearly options to try. It should have a statblock - but it should also have listed motivations and assets and everything else the dm needs to roleplay the creature set up (or at least suggested) ahead of time. Some might lend themselves very heavily to a certain approach (you can't negotiate with golems) but you should also plan in terms of multiple means of interaction.
I got the idea form the intro to Zipperon Disney's YouTube video on exploration. It's a throwaway line, but it stuck with me.
(The downtime pillar wouldn't be a pillar on its own - it would be set up in relation to how it can support doing dungeon stuff.)