As I understand: * Mercator has accurate shapes but inaccurate sizes. * Equal Earth has inaccurate shapes but more accurate sizes.

Inaccurate shapes are not a problem when looking at a big (global) static picture. Your brain still matches the shapes and connects the dots. However in a digital app when zoomed in, street views of NZ, east China, Chile, and Canada would be completely warped to be borderline unusable for most people. Zooming in would likely force the app to switch or gradually transition back to a shape-preserving projection, likely Mercator.

If Google would actually get on board with this UN resolution, would we start seeing shapeshifting countries and cities in Google Maps whenever we zoom? How do you counter this?

You can smoothly interpolate between different projections, and the scale differences are large enough that it should be gradual enough not to notice much. Either way, globe view is always superior.