No one is denying visual differences. It is more a matter of how one approaches that evidence. If TOS is the only benchmark for starship design, then DSC, and even the TOS films, fail on that regard in maintaining continuity.
I’m not denying visual changes either - nor am I saying anyone else is

simply that, as you say, it’s how one deals with those changes.
As for DSC failing based on TOS alone I agree - for me the whole aesthetic update, although necessary, could have been more subtle and closer to TOS.
The TOS films is more murky I think. I see why you’re saying that it’s a continuity fail based on TOS alone as it is a drastic departure from TOS. As is DSC...
But, on a fundamental level, the changes from TOS to TMP have more similarities to them than DSC to TOS. For instance: tactile controls on the consoles and no (observed) touchscreens appear on both pre and post refit eras. Viewscreens on the bridge, not widows also occur. Displays that simply look like flashing lights to us, but presumably signify some complex operation of the ship in-universe (requiring the viewer to use their imagination). A relatively small bridge and a feeling of it being a well lit and functional workspace. These are all things that are carried over from TOS to TMP. It feels like the same ship/universe on a fundamental level.
Compared to DSC’s updates to make it look more modern to us (despite the fact that we actually have no idea how the Enterprise consoles worked in the 60s and they may well have been holograms that looked like jelly bean buttons that were multifunctional and reconfigurable), it’s as though some of the fundamentals have been changed. More touchscreen consoles, hologram communication with sophisticated tactile interfaces that can interact with surroundings lightyears away (echoing Rimmer’s hard light drive in Red Dwarf). Windows at the front of the bridge. Much larger bridges that look like they’re more about style than substance in their design - the larger distances between colleagues mean that communication is less straightforward than it once was. Complex LCARS graphics that bear no resemblance to the simple displays of yore, showing little reverse engineering from the TOS look to reflect the situation 10 years prior to TOS - an iPhone from 10 years ago would have fundamentally the same software as a modern iPhone. As would windows look fundamentally the same 10 years ago as it does today.
Based on this evidence, I don’t believe that DSC is set in the same universe as TOS, but I have a much easier time believing that TMP onwards is set after TOS.
However, the story is more than just the visuals for me. It's a matter of the characters and the events as well as delving further in to details that TOS simply could not do.
I totally agree with this. I hope that the new writer situation gives us better stories in s2 as s1 didn’t live up to my expectations story-wise. But I remain cautiously optimistic. Every Trek show has had to start somewhere.