What problem does this feature solve?

currently, extends component missing slots of child components example: https://codesandbox.io/s/vue-extends-component-with-slots-not-working-3niol?file=/src/App.vue

What does the proposed API look like?

added child's slots to parent component

This is expected because extends is like mixins as doesn't reuse anything from the template as it has been discussed before under the name of code blocks and others in vue repo. You need to forward the slots yourself