Why can't I enter my password on Apple devices running iOS 13 or 14?

Trick question! Although it is not visible, the password actually does get entered when you type into the password field.

However, from iOS 13 onwards, as part of its security enhancements, Apple blocks the streaming of the password field, and hence is not visible while it is being entered.

For a successful Sign in, finish entering the complete password as you normally would, even if it doesn't reflect on the screen, and then click on Sign in.