I felt sufficiently invested in Burnham's stakes that moving in to the epic stakes was more seamless for me.
I know that will vary since Burnham, as a character, is not someone everyone connects with. For me, the Klingon and Mirror Universe arc all tie back to Burnham and so the personal extended to the epic for me.
I'm not really making an argument about the effectiveness of the personal stakes, so much as the epic scope of the story not being require to tell the very personally-focused story they set out to tell. Because - as I said - most of the issues that people had relating to canon were because they decided to put so many "first ever's" in the show. Off the top of my head, Michael was "Starfleet's first mutineer," was the first human to ever attend the Vulcan Learning Center and Vulcan Science Academy, served as XO under one of the most decorated captains in Starfleet history, got a chance to serve post-mutiny on the most technologically advanced ship Starfleet had ever made, which could travel anywhere instantaneously, was personally responsible to the deaths of two nominal heads of the Klingon Empire (T'Kuvma and Kol) and personally installed a third one (L'Rell). While her foster brother, Spock, is no one of particular importance yet within-universe, there's evidence that Sarek is more important than just being a senior Federation diplomat, considering how he holds considerable sway in Federation governance based upon his discussion with Admiral Cornwell.
Now, let's step back and consider how you could "de-epic" all of this. Micheal Burnham serves as first officer onboard the U.S.S. Shenzhou under Georgiou - a fine, but not quadrant-famous - captain. There is an encounter near the neutral zone with the Klingons, where a bad judgement call she makes (not "mutiny") results in her captain and surrogate mother figure getting killed. She is court martialed and ejected from Starfleet (not imprisoned for life). The situation in the neutral zone heats up into an outright war, and Michael blames herself for it - even if Starfleet does not directly. The war is going badly, and Starfleet is desperate for warm bodies, so she's given a second chance, serving on the U.S..S. Discovery - a third-rate tin bucket - under Captain Gabriel Lorca. The main story flows out from there, with the same main cast as Season 1, with identical personalities. though obviously the episodes themselves differ in some details. I would personally end the season with something similar to Michael's encounter with Kol in Into the Forest I go, because in terms of her character arc, it was a nice closing - she came full circle, faced down her fears related to Klingons, and won back Georgiou's pin.
The crew of the Discovery were no more integral to the survival of the Federation than the Enterprise was during the TOS run. There are numerous TOS stories where the survival of the Federation is at stake and only the crew of the Enterprise are there to act.
The difference though is in TOS there was never really any indication that Kirk was anything other than a young hotshot captain. He mostly got grief from admirals, and we didn't see any award ceremonies until the movies (when the legend of Kirk had been built up in fandom).
Basically, when watching TOS, it's easy to imagine there are dozens of other ships just like Enterprise, out there having identical adventures - sometimes saving Earth in the process. It's just we never see or hear anything about them. But in Discovery, it's hard to believe that there is any other ship which is as integral to solving the issues of the day - because we're explicitly told that the ship is special.
You appear to not have read much science fiction or watched much science fiction. In science fiction, the big picture and small picture are often best utilized when they are intertwined. Science Fiction is quite a bit different in many ways than the amorphous 'fiction' you speak of.
Actually, I almost exclusively read science fiction when I do read fiction. Lot's of science fiction books have relatively low stakes beyond the personal, and are very compelling. For example,
Rendezvous with Rama, The
Left Hand of Darkness, Flowers for Algernon, Solaris, and
Do Androids Dream of Electric Sheep are all examples of science fiction books where the Earth/galaxy/universe isn't really under threat, and they are all very compelling in their own way.
Of course, there are examples of epic stakes in science fiction. I love David Brin's Uplift Universe novels, and most of them revolve around whether mankind will continue to exist, and involve galaxy-wide conflicts by the end. They work though because of the attention paid to character. On the other hand, someone like Kevin J. Anderson, or even Peter Hamilton, does epic without really compelling characters, which makes the work far more tiresome. I'd also note that I really think in literary series you can only build up to the big global stakes once (even if across an eight-book series), and then it's done. Again, if you have the same friggin protagonist save the Earth/galaxy/universe from different threats again and again, it just starts feeling like pulphouse, or a comic book, not a serious work of fiction.