It's not just the old web. I have a small website that does not show up in Google's results. When searching for the title of the site it's the 7th result on DDG yet does not appear on Google at all (Google runs out of results after 19 pages, which is also ridiculous considering the search is very common). When I search for the domain name (sans tld) on Google the associated YouTube page is the first result, also an unrelated Twitter account, and even a Reddit comment that links to my website (Reddit is a more effective search engine!). But my website is not listed. Finally, if I search for the full domain name the website shows up as the first result.
This has been a problem for three years, it's incredibly frustrating, and also demotivating.
Google does have some webmaster tools where, after you prove you own the site, you can check if there's something you have to fix about it to get listed, can submit new URLs for them to scan, etc..
This has been a problem for three years, it's incredibly frustrating, and also demotivating.