I don't want the blog's home page have H tags applied like on the posts' pages.
That's the cause of the problem as originally theme makers started from the home page. There, you have your blog's name which is H1 and then one or more posts where H2 or H3 is applied to their titles.
Post pages are different story.
That's why two headers should be provided: one for a home page and another one for posts pages.
I wondered about what people actually do to make this right.