I do quite a bit of site migrations (rebranding, domain changes, redesigns). Usually the goal is to preserve (and improve) organic traffic as well as preserve page authority of old pages (and transfer it to new pages).
I see that your case is different because your two sites are not related. And your first site hasn't had a chance to aggregate links (authority) yet.
The short answer is that your current situation should not affect SEO of the new site. "404 or 410 Status Codes Will Not Impact a Website’s Rankings: If Google identifies 404 or 410 pages on a site, it will continue to crawl these pages in case anything changes, but will begin to phase out the crawling frequency to concentrate more on the pages which return 200 status codes." Source: https://www.deepcrawl.com/knowledge/hangout-library/4xx-errors/
However, here's few things you can try to simplify Google's life in understanding your site:
1. Verify your domain with Google Search Console (GSC)
2. Create and XML sitemap and submit in GSC
Although "site:" is a good quick lookup option, Google suggests to use Coverage report in GSC to get more precise and reliable information on the state of your page indexing. Keep an eye on that report and don't stress too much about 404s in SERPs. They'll get removed eventually. It might take up to 12 months for Google to remove those completely but in most cases it happens much faster.
I hope it helps. Feel free to book a call if you have more questions.