Backport the new validation mechanism to Vaadin 14

[vursen]

This ticket is created for tracking the progress of backporting the new validation mechanism to Vaadin 14.

For any web component PRs related to the backport, please reference this ticket.

PRs to be backported

• Ensure no initial validation when the field has an initial value and invalid #4150 (canceled as we cannot make breaking changes in v14)
• feat: fire a validated event on validation #4084
• refactor: add methods to make setting invalid overridable #4097
• feat: add internal has-input-value-changed event #4276
• fix: sync the hasInputValue state on input #4335
• fix: check for bad input when updating _hasInputValue #5092
• refactor: move input value getters and setters to InputMixin #5586
• fix(input-mixin): reset _hasInputValue on programmatic value clear #5599
• test(clear-button-mixin): ensure _hasInputValue is reset on clear button interaction #5609
• fix(date-picker): reset _hasInputValue on programmatic input change #5625
• refactor: simplify _hasInputValue workaround, improve JSDoc #5645
• fix(time-picker): update _hasInputValue on programmatic input change #5638

Additional context

Related to vaadin/platform#3066