> This requires having the resources to support developing two such products at the same time using separate teams for each, which is not something many companies can afford.

This is definitely the ideal situation, but as you said most companies do not have the resources to greenfield a separate project. Additionally convincing the higher-ups that its worth it is a difficult task.

> Another good way is to rewrite piece-wise, in place (refactoring). There are lots of different ways to do this regardless of architecture, ultimately you're essentially creating a bubble of new code that grows and grows until it completely replaces all the old code.

> Work component-wise in an "add + deprecate" fashion.

Everywhere that I have worked this has been the route we've taken. We isolated the old behavior until we were able to fully ween clients off the product (or perform a data conversion and completely use the new code).