In my view yes, you should fix the issues. The key about good seo - is not "only pulling certain levers" but where feasible doing everything right. We find when we do all the basics right the results sometimes take care of themselves. It lets you sleep easier as well. Instead of thinking "what if?".
H1's influence rankings - I have implemented some that have had a small impact, however, most do not. There is impact enough to optmise. The H1 is also great for usability, which can increase conversions. This is a ranking factor - time on page, bounce rate etc. It could increase your quality score if using Adwords. Additionally if a customer clicks on a search result which has a consistent message via URL, Title tag and H1 it is confirmation to click-through to a page and builds trust on topic - hopefully good content follows.
For the visually impaired marking up your HTML with header tags is also appropriate to facilitate screen readers.
I would also review your URL structure. If you want rankings then they again in my view should be consistent with the page offering.
Hope that helps.