primers are "non informative"
I'd agree on that, however my reasoning is a bit different. (Traditionally) you target the conserved regions, so there's close to no variation in that part of the sequence anyway (except the designed ambiguities). What you don't know is how much of the sample you didn't amplify because your primers didn't match in the first place
primer sequences in your fastq is not (always) the same as the real biological sequence
Out of curiosity, is this personal experience or can you link to some systematic observations?