Seriously - how many times did Kirk actually save the Earth in TOS? There's time travel stories - most notably The City on the Edge of Forever, and arguably Tomorrow is Yesterday and Assignment: Earth as well. There's the terrible The Alternative Factor, where the stakes are "the entire universe." But besides that? Zip. There are cases like The Doomsday Machine and The Changeling where the Enterprise defeated threats that might - eventually - threaten Earth. But if Kirk & company failed, there might have been another crew waiting in the wings to pick up the pieces. And in most cases, the stakes were smaller than that - it was a planet of the week unrelated to the Federation, the survival of the ship itself, or even just the survival of the main cast.
Um The Changeling wasn't a threat that 'might' one day show up. Per NOMAD: "I shall return to Launch point Earth. I will sterilize." <--- As NOMAD had found it's creator, it was pretty much a given that next stop...Earth; and with the ability to trow energy bolts at Warp 15 - it could get there fast.
It seems you downplay everything that doesn't fit the narrative of "ST: D tries to be too epic." Also, the show pretty much had the entirety of Starfleet involved in a full scale war. Yes, the Discovery went to the Mirror Universe for a few episodes and took out a ship that was damaging the Mycellial Network and could have ended up ending the Universe eventually - but hey, like you admit, even Kirk and Co. saved the Universe once in TOS S1 - "The Alternative Factor".
Now ST: D S2 went the epic "save the Universe" route whole hog - yep; but hey given in brought in the 1701, Captain Pike, Spock ane Number One to do it -- I'm fine with that. Here was a situation that was epic just because of teh personalities BESIDES Burnham involved.
I think what throws TNG fans off is the fact that overall, TNG WASN'T an 'Adventure Show' in the same way the TOS era (TOS and TAS plus the films and JJ Abrams take on TOS) was. Yes, TNG had the occasional 'Adventure' episode, but compared to TOS, TNG was mostly Board Room conferences of exposition, with Picard lecturing and sermonizing more often than not.
Personally one of the things I like a lot about ST: D is the fact Star Trek is back to having 'Adventure' back as a big part of the mix.