My site isn`t using H1s at all

If a site isn`t using H1s at all and is using other headers in a odd way e.g. H2s all say "contact brandname" how hard would you push to change it? Given I expect some pushback on how important it is as there will be design work needed (minimal though I`d imagine). I`m thinking it`s v important to use H1s and other headers properly but a sanity check that you guys agree would help me. Thanks, Kath