Good question about 141.

121/135 is required to keep and report specific records.

Part 91 employers (includes 141) are required to return only those records which they happen to have. A 91 employer might not keep any at all (legal), but if they do they have to return what they have if they are the types of records required by PRIA.

141 employers do have to keep some records related to standardization, but that kind of thing is usually a formality so those records are usually not a problem unless the CFI was a problem child to the point where he got removed from duty (in which case he would probably have been fired anyway).

A potential issue with 141 is that in many cases the CFI also trained there as a student. I've seen multiple cases of 141 employers returning both employee records AND student records for a CFI who also trained there. Not correct or legal, but the min-wage girl who answers the phones and the mail might not understand the distinction. So in addition to your spotless 141 employment record, an airline might get your 141 student record including all incomplete stage checks, repeated lessons, etc.