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When I do site: on Google for my website, the homepage is nowhere in search results. In fact, I can't find it anywhere at all.
If I search my brand name, it comes up and the pages are indexing. My website is indexed overall and I can't see anything on my site that would be blocking Google from crawling the homepage.
Will this have an impact at all? Is it something I should be worried about?
No, this will not affect your "whole-site-seo". At least not directly.
The ranking of your pages within googles sitesearch results (site:yourdomain.tld) especially for newly indexed sites is affected by your internal linking structure.
You could describe it like that: "What gets linked most (internally), is most important for google." (Although there is more behind it, but for starting up, this is enough to know)
Just make sure everything is linked well and tell the robots to index your homepage.
Have a look at your most important pages, link them internally well and smart - this should be a good thingy for starting.
site:search results to make it harder to reverse engineer their algorithms.