SevenCubed - 4:43 pm on Sep 20, 2013 (gmt 0)
All my sites have H1 H2 H3 H4... used properly and don't suffer at all in SERP.
And like you said -- don't be rushing off to trash structure, for many reasons, not the least of which there are many search engines that use them.
If google doesn't that's their choice, they can just ignore them if they are on the page.
I think the low density of word repetition is more likely to be the deciding factor rather than the absence of H1 tags.