Is Piaget’s stage theory too limited in its scope? Can Piaget back up his theories other than to provide examples of his children?

Jean Piaget's theory of cognitive development may be deemed "limited in it's scope" in a modern sense, but it was considered a rather innovative approach to understanding developmental psychology at one point in time (not to mention how it still serves as a foundation for our understanding of cognitive development today).
Modern psychologists still use (and generally accept) Piaget's stages; however, cognitive development is now widely viewed as a spectrum rather than abrupt stages children pass through on their way to adulthood, which is precisely what Piaget suggested up until his death.
Countless studies have been conducted on the grounds of Piaget's research, most specifically in regards to childrens' classrooms and the nature of knowledge being learned/acquired by one's environment rather than biologically instilled within each individual.
Piaget conducted plenty of studies himself over the course of his lifetime; although most of his research does require critique/skepticism on the part of controlled variables and technique, most of it did lay the groundwork for future research (a lot of which is still being conducted today).