Disclaimer: Wasn't sure if this fit here or Little things since I neither know how common this is nor how significant most consider it.
I think despite the better scale of the models on screen vs Sprites and some improvements to the team writing, ORAS has a decidedly inferior atmosphere to the Legendary encounter in the Cave of Origin compared to RS. I would even go as far as to place those above how the Emerald resolution is handled, though that one is down to preference moreso than ORAS I feel doing "the same but weaker".
Riding on the Pokemon to the battle site before they Primal Revert is a little silly to look at, and I also think it kind of breaks the pacing/tension. RS it's a simple descent through the cave, rumbling each floor before you reach the last room. The music cuts out, the Legendary gives one cry as you approach, then the Orb catches its eye and the music kicks in for the encounter. I think it epitomizes "Less is More" in presentation, giving the Mascot a presence befitting the escalation they had this gen from "Large creatures" to Titans bordering on Deities. It gives this sense you're only within their notice because of the Orb even.
ORAS I think weakens it a bit because it just feels "safer" ironically. They give you this protective suit and the Titan is clearly aware of your presence, bringing you (actively or just as a consequence) to what is essentially an arena as if accepting a challenge. Maybe there's a nostalgia to it but it makes them seem like "just" big Pokemon despite the Primal lore existing to elevate them back to a monstrous status (ironically this happening when they held on pretty well against the God-Like Ubers that came in the intervening Gens)
I'd agree with this.
Since you mentioned Emerald, I've always held that Emerald is the best iteration of the story and I think part of what I like about it is that the player character is important to the story, but not integral - sure you awaken Rayquaza, but there's nothing inherently special about the player that enables them to do that. It's more that everyone else is too preoccupied to do it - Wallace unlocks the Sky Pillar but then hurries back to Sootopolis out of concern once the weather distortion spreads, Archie and Maxie are fixated on trying to control Groudon and Kyogre (apparently it never once occurs to either of them to try switching orbs for a second), and Steven... well, actually, Steven could have done it but let's ignore that.
But it's not anything inherent to the player, beyond their generic skill as a trainer, that makes the plot happen. They're not the chosen one (which I think totally works in other contexts - BW or HGSS for instance) nor do they have any special bond with Rayquaza or any of Hoenn's other legendaries in the way they're indicated to have in ORAS. In fact, it's one of those stories where if the player wasn't present things would largely probably turn out the same way - someone else could have defeated Team Aqua and Magma at Mt Chimney, and Rayquaza ends the fight between Groudon and Kyogre no matter who wakes it up. The player's role is important but not entirely irreplaceable.
All of this is to agree with your overall point that ORAS making the story feel "safer" lessens it slightly, because you get the sense that it'll turn out fine - Groudon/Kyogre is perfectly willing to wait and let you challenge them. In RS there's more of a sense of urgency, that you absolutely must catch/defeat them before the damage they're causing goes too far. In terms of tension and pacing it's definitely superior, and Emerald improves on that by making the civilians in Sootopolis actually give a shit - they're all assembled outside, looking on with concern as Kyogre and Groudon clash.