Why did my PDF show "no form fields"?
The document is a plain (or scanned) PDF without embedded form controls. Only PDFs authored with interactive fields can be filled this way.
Should I flatten the form?
Flatten when the form is final. It prevents further edits and renders identically everywhere. Leave it unflattened if the recipient still needs to change values.
Is my form data private?
Completely, the values you type are written into the PDF by pdf-lib in your browser and never touch a server. That matters, because forms are usually full of personal data.