Regulatory regions make up about 2% of the genome, so the average neuron has about 30 mutations in regulatory regions by the age of 40. So if we can keep our de novo mutations from increasing that number very much it will probably be ok.

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 

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