If you unintentionally insert a few base pairs into the promoter region of some protein, will the promoter just work a little less well, or will it break altogether? I
Molecular Biologist here. Promoters (and any non-coding regulatory sequence for that matter) are extremely sensitive to point mutations. Since their sequence determines how well the RNA-polymerase binds to them, any change in the sequence of bind motifs or even in the distance between these motifs has a major (generally negative) impact in transcription initiation efficiency. https://www.nature.com/articles/s41580-018-0028-8
In fact, there is a whole field of research based on randomizing certain parts of a promoter to create a library with different properties/strength.
Well known library for bacterial promoters: https://parts.igem.org/Promoters/Catalog/Anderson
more info on promoter libraries: https://sci-hub.se/https://pubs.acs.org/doi/full/10.1021/acssynbio.8b00115
I agree that in the grand scheme of things it would probably not make much of a difference. Also your 2% estimation is generous, if you consider that in any differentiated human cell most of the genes are inactivated. Mutations on those genes would thus be harmless