No. ST: D does not overwrite ST: TOS. They are two separate shows with differing continuities.
You're misunderstanding what the word "continuity" means. It doesn't mean that every last detail fits perfectly. It means that the stories maintain the pretense of being continuous with each other, even when inconsistencies of detail exist. The current Marvel comics asserting that Tony Stark built his first Iron Man armor in Afghanistan or in some made-up Mideast conflict are in the same continuity as the 1960s Marvel comics asserting that it was in Vietnam, because the overall
events are treated as continuous even as the details are adjusted. Similarly,
Discovery is in the same continuity as TOS and TNG because it directly references "The Cage" and "Unification" as events that happened in its narrative universe. Changing the visual depiction or the surface details does not make it a separate continuity; it just makes it a different artistic interpretation of the same continuity.
This is fiction, after all. It's just pretending. So you can pretend two things fit together smoothly even when they don't. You can pretend that what happened one way in an old story "really" happened a different way instead. Continuity is as much a part of the illusion as anything else. As with every other aspect of fiction, we know it isn't actually real, but we choose to suspend disbelief about it and accept the pretense.
The ship, uniforms, and tech in ST: TOS are how they're suppose to be in ST: TOS's continuity.
The ship, uniforms, and tech in ST: D are how they're suppose to be in ST: D's continuity.
Same continuity, different styles of representation. In the same way that the
Star Trek and
Star Wars animated series are in the same continuity as their live-action movies and series.
Clone Wars Anakin hardly looks anything like movie Anakin; he has a very different face and voice, even aside from being a cartoon instead of flesh and blood. But we understand and accept that the two very different portrayals represent the same character in the same continuity. They are
narratively continuous even though their superficial details are discontinuous. It's the same with different portrayals of the
Enterprise or the planet Vulcan or Pike or Saavik. Continuity is about story, not imagery.
Both versions may be available on Blu-ray, BUT, some of us don't have a Blu-ray device or the discs for one reason or another. We have to rely on terrestrial TV which only shows the "Remastered" version. So the often touted claim that one can easily choose to watch the non-"Remastered" version if they want to is incorrect.
Nothing is available to everyone, but that doesn't mean it's been destroyed or replaced. I can't afford to go to the Louvre to see the
Mona Lisa, but it's still there to be enjoyed, and if someone paints their own new interpretation of it, it doesn't affect the continued existence of the original.