I have progressed further down the path and I am documenting my findings here to perhaps help some other poor soul who is travelling down the same path. I am not an ESRI expert, my findings are based purely on what I see.
I had previously said we were dealing with polygons, to be more precise we are dealing with polygons and multipolygons.
If you have non-ESRI data and want it in an Enterprise geodatabase (and nowhere else) then using FME with an input of SQL data, passing through an FME Geometry-Validator is a reasonable way to go.
If however, you then take the resultant Enterprise Layer and move to a File Geodatabase you may well find additional 'Self-intersections'.
On the other hand if you want to be able to use the data in both File Geodatabase and Enterprise geodatabase then the easiest path is to use FME to take the non-ESRI data and output it directly to a File Geodatabase (do not use FME Geometry-Validator as it produces many false positives). Once it is in the File geodatabase you can then use ESRI Repair-Geometry to fix the self-intersections (hopefully). The resultant File Geodatabase layer can now be copied to an Enterprise geodatabase with confidence (an added bonus is that it can be exported to a SHAPE file as well).
My overall conclusion (if someone more knowledgeable can confirm) is that the File Geodatabase is more aligned with a SHAPE file than an Enterprise geodatabase. Perhaps the accuracy that a Fiile geodatabase stores its data is the same as that used for a SHAPE file?