So, this is kind of all over the place, but anyway:
Should Games take themselves seriously?
There's plenty of room for the serious and the silly across all media. It doesn't have to be one or the other.
Personally, I think that a lot of games these days are losing sight of the fact that a game doesn't have to be realistic in order to be immersive.
Don't really see such a trend. There's trends in gameplay, sure, but not really trends in storytelling outside perhaps more focus on multiplayer-only title where story/lore is found outside the game itself.
Take the difference between Doom (2016) and Borderlands. Totally different games I realize, but despite Doom's minimal story and focus on gameplay, the story feels real and urgent(argent hur hur) because the characters, namely Samuel Hayden, take what's happening very seriously.
Borderlands on the other hand does the opposite, being full of pop-culture references and jokes about "leveling up" etc.
So, I haven't played Borderlands, but of what I've seen/read, it does seem to be the case that Borderlands is very tongue in cheek and whatnot.
On the other hand, I disagree that Doom 2016 takes itself seriously. It doesn't help that the story is terrible as a whole, but the lore is thrown together in a haphazard way. The Doom Slayer doesn't care about what's going on, and there's the unspoken expectation that the player won't care either bar "yeah, there's some lore here that we threw together, read it or something." If you want an example of a Doom game that does take itself seriously, look at Doom 3.
To my knowledge nobody cares about the world or characters in Borderlands, they play it for the gameplay, while there are many people who are at least somewhat invested in the world of Doom in addition to the game.
For Borderlands, world no, characters yes - Borderlands is clearly a character driven game as far as plot goes. As for Doom, I've barely seen anyone invested in any aspect of its story.
I mean, there's potential in Doom 2016 for story. I've written more for Doom 2016 than I have D3 for instance, yet that's still dealing with the crumbs Id Software provided. But what's provided is done so terribly that I've barely seen anyone give a shit.
I think Starcraft and Brood War feel a lot more serious than Starcraft II. And I think there are plenty of other examples of this.
For argument's sake, let's take that as being true - I'm not sure what that means. SC2, even if it's less serious than SC1, is still taking itself seriously. There isn't some Borderlands/Doom divide here.
I mean, I agree with you that if the setting is taken seriously in-universe, it will usually be taken seriously out of universe, but you can easily get invested in elements of setting even if the setting isn't serious for instance. Or, the setting can be jovial but still be taken seriously. Overwatch is light-hearted for instance, but its setting is taken seriously in as much that it's self-consistent, and that's reflected in stuff like wikis and ff.net, where you don't need to look hard for stories set in the Omnic Crisis for instance.