I would like to know if FME has a reader and writer that directly read and write data back to the ArcSDE database.

It does, yes. Granted, you'll need an Esri license. You can find more details here:

https://docs.safe.com/fme/html/FME_Desktop_Documentation/FME_ReadersWriters/geodatabase/geodatabase.htm

See also https://community.safe.com/s/article/notes-on-fme-and-esri-versions-and-compatibility

@fme_can_do_it​ , FME does include an ESRI SDE reader/writer that you have access to dependent on licensing.

Here is an article going over the various ESRI readers and writers.

https://community.safe.com/s/article/comparison-of-fme-readers-and-writers-for-sde-and