This is a thing wich is pretty much normal for a lot of CMS-Templates. It is not best practice, each SEO-Tool will tell you to use only one h1 wich is unique. Don't have equal H1-Tags everywhere. Thats the best practice.
According to John Mueller and my experience, Google is not stupid. It is possible to have more than one H1-Tag. Now it depends on how much factors you are serving well. You and your competitors. These H1 alone is not such a big factor. It is working if you have a second headline with the main-topic of your website. A lot more stuff is relevant. If it is not easy possible to change it, you can deal with it and care about other factors. Special if they are in different sections.
If it is possible, I would allways follow best-practices (specially for smaller or newer domains/companies). One thing sounds wrong "he H1 text being the brand's name, as this is what we'd want the title of the page to be" - I mean, the title (bet you ment the h1) should be what the page is about. Without alt-attribute you current H1 is a simple image, wich means it is empty.
You ask what is working better - Better is to have an H1 unique on every page. Dont use Headings as style Elements and yes, give an alt-attribute for your logo. Thats defenetly a better way, but it is not impossible to do like you did. Like I said, I am pretty long in this business, thatswhy Logos, Sidebar- & Footerheadlines, all style Elements are not SEO-Elelements when I create Webpages. But yes - I also work for websites without changing h1-logos and h3 menu-items (cms-reasons). And it is possible to rank in hard topics. Favorite answer - it depends.