It looks like the issue may be related to a customization for displaying H1 tags in version 0.1 of the theme, which may no longer be supported in version 0.2. I recommend reaching out to a developer to help resolve this.
H1 tags are normally on the collection pages. looks like someone did a lot of custom work to the site. someone needs to fix on the technical seo in a big way. there are major performance, best practices, accessibility and seo issues that any developer should have addressed when creating the site