This is not particularly a GAM thing, although I suppose it could be. I have a form that updates a sheet, but when a form response is added, the sheet's modifiedTime does not change.
This is a recent behavior since July 20, 2023. Before that, form responses did update the sheet's modifiedTime.
I am using
gam user myusername print fileinfo myfileid
to get the modifiedTime of the sheet.