I think Google Maps already does this at least
It clearly doesn't - you can zoom out and see this for yourself. It uses an equatorial variant of the Mercator projection.
https://i.imgur.com/D6fMLop.gifv (Native vid: https://i.imgur.com/D6fMLop.mp4)
As for whether it looks "so bad," it's hard to tell from that video, because it's a small segment of the map, all in the northern-most latitudes.
The Mercator projection is very shape-conforming (except close to the poles, as you can see from the huge top of Greenland), and its size distortions depend on latitude, so locally it's always going to be very "good" compared with a globe. You're not going to see any huge "errors" just comparing the before-and-after shapes of Greenland (although, again, the top of Greenland is very distorted there), or comparing its size to its closest neighbors.
The first frame is the Mercator projection; the whole conversion (See GP: "..I think any digital map that lets you zoom out more than a hundred miles or so should be an accurate 3D projection" and then the reply "I think Google Maps already does this at least") is about now on GMaps you can enable "globe" option, which uses a more accurate 3D projection.
The video shows before/after enable globe option.