Don't want to keep harping on about it but just as an example (names & location changed, as I haven't asked him if I can tell the tale)
I was asked recently by a pizza restaurant owner in Melbourne Australia, to take a look at his site with regard to updating & getting his business back on track.
after looking at everything initially, I changed his h1 header (which was the name of his pizza place) to a H2 header.
I then placed an H1 header lower down on the page, below the navigation, & the full-width image (but still above the fold) this H1 header text " Caco's The Best Pizza in Melbourne" (Caco's not the real name).
Within 2 weeks of just doing that (not even re-designed the site) his pizza place had crept back up the SERPS 3 places and was back on the first page.
It's not exactly brain surgery, but if you think about it, what would someone type into a search request for a pizza restaurant in Melbourne, well for a start they wouldn't be typing the NAME of his restaurant unless they knew it already so likely they would be asking Google to show them the sites for "The Best Restaurant In Melbourn (or words to that effect)
By placing those words into an H1 header even it is below the H2 header on the page, & the fact that the H1 was a "section" header, & therefore the content of that section, was given more importance &relevance that just being a site title; (back in 2010, it was H1 had to be the first, thing followed by H2 etc)
So consider this
People who ask Google to show them a website for serving legal papers in the Denver aren't likely to type in your company name because if they are searching then they don't know it (if they know it & what you do, they would come direct or use bookmarks) they will ask a question.
So do some investigative work, find out what the popular questions most asked of Google are, and incorporate the answer into your H1 header (which as has been shown doesn't have to be the main title anymore.) Putting your company name as H2 header (which will still get some ranking score) will not hurt rankings
The h1 header is the title for a section of the page relating to the relevance of your site & services, but then again there isn't much said in the sections'
As a final aside,
Google ranking is only 1 of over 200 things they look at on a site with regard to giving their customers
your visitors the websites that Google thinks will show their customers
the best results(answers) & visual experience they can get.
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