Once again, Greg Cox puts it well.
As somebody who (dimly) remembers watching TOS on its original run on NBC, I'm quite enjoying DISCO. It's the show I wanted ENTERPRISE to be: a bold attempt to reinvent STAR TREK for the 21st century.
The visual upgrades don't bother me nor do I require an in-universe "explanation." Sometimes a change in art direction is just a change in art direction and it's not as though Trek is a historical documentary. I don't expect a TV show in 2020 to zealous recreate the look of a 1960's TV pilot.
And as for the Klingons . . . well, if we survived the Klingons getting a major makeover in 1979, we can survive some new, new Klingons now. There's no such thing as a "real" Klingon.
I am an obsessive TOS fan, who later came around to liking and loving other parts of the franchise. I've been a fan since TAS first showed up while I, a seven-year-old, was busy wolfing down sugary cereal on a Saturday morning. Watched TOS daily on WPIX-11, and grabbed any kind of additional media that I could grab in the 70s. I enjoyed TNG during its run, but it took the lead up to the 2009 movie to get me interested in finally watching all the rest of the franchise, especially after the lackluster TNG movies.
I'm always ambivalent about having to produce those creds, like it's some sort of card I need to show, when I say I love DSC! I think because TOS is "my" Trek, I am predisposed to prefer the stuff that's TOS-adjacent. It was probably a smart move on their part to put it in that setting, as I'm not sure I'd jump to watch the show in the setting it's going to now. But now I love these characters, so I'm aboard for the ride.
I find it perplexing how downright angry some people of my generation get about the continuity inconsistencies in DSC (though, I've generally observed it's more the TNG generation viewers that seem to get more wound up about it). TOS was routinely inconsistent with itself, and my experience is that people, while pointing these out, would generally try to make up ways to make it work, for fun, instead of going ballistic. Generally.
You'd think that anyone who grew up with Gold Key Comics, Power Records, and the Bantam Novels would be less disposed towards getting their shorts in a bunch about how Klingons look (I don't recall anyone storming out of viewings of TMP in 1979), or previously unmentioned siblings (hey, it's Sybok!) showing up. You'd think that anyone who has seen Gene's Vision-tm tinkered with so much between 1979 and 2002 would be more chill about a new iteration of the show.
For me it boils down to my enjoyment. I regret I didn't watch ENT back in the day because somebody convinced me it had to look like Forbidden Planet. It's now one of my favorites. I was determined to give DSC a shot.
And I liked it. I've found it the most consistently entertaining first two seasons in the franchise since TOS. TMP and on, I never lasted much more than one season during their first run, and only came back to them later (DS9 is probably my favorite of the era). DSC, I've been hanging on every episode. It's not perfect, but it's fun: and that's what got me into Trek.
ETA: Oh, and about Picard. I've been finding that I'm getting emotional about seeing the return of characters whom I was not that emotional about in their first run. I'm enjoying it immensely, although I'm finding it less daring than DSC, which is probably intentional, given how prickly TNG fans have been about DSC.