actually when you type into google "google maps projection" it always says, that they are using mercator or some hybrid based on mercator, until you "going into space". Good looks does not mean automatically that it is handy. I worked before for a map printing company and other world projections that are on paper or screen oval, bringing in the problem of filling "holes" in the corner. So if you are solving this problem that exists since maps/projections are existing you are going to be a rich man.
BTW (state of google on the 15.09.2019):
google maps (desktop) starts you usually in mercator and switches then to "globe" mode if you zoom out. I guess they "cheat" with mercator because the want to save bandwith & computing power. This is why monstrosities like google earth (and pro) existed, this programs/apps are for pro users.
google maps (mobile)still ueses some kind of mercator projection.