The article in the OP does not seem to understand how websites worked. This is not unusual for news sites.
Let me make it simple:
- You buy a domain name, which is how users know to translate neogaf.com to a particular server on the internet. GoDaddy, Network Solutions, NameCheap, Google Domains are four of the most common examples of companies called registrars, which sell domain names. ICANN, listed in the OP, is like a standards organization that helps those companies coordinate and sets rules for the sale of domain names. If you lose your domain name, you will need to buy another domain if you can get one.
- Typically if you're at all large or privacy oriented, you have a CDN or reverse proxy. This is something that sits between users and your actual server for the purposes of protecting your from DDOS, anonymizing your location, etc. CloudFlare is the most common example of this. If you lose this, nothing happens in principle, although it'll make it easier for people to bombard your server, hack you, or identify your host.
- The actual data for your site is located on your web server or servers which you have with a web host. Sometimes the host has one physical server hosting multiple sites (shared hosting), sometimes it's one physical server with multiple VMs ("Virtual Private Server"), sometimes it's one or more dedicated servers you're renting from the host, sometimes it's servers you physically own and store in a host's data center ("colocation"). If you lose this, you have no more website, and depending on what the host does, you might lose your live data. There are many of these hosts.
Losing your domain cannot "control if you delete your site". The site is on the web host. Some registrars also provide web hosting, and if Stormfront did that, then their data might be held hostage as a result of suspension of the domain. But it is much more likely that the site and data itself is fine, they merely do not have a current domain name for you to access it. To actually take the site down, you would need to seize the domain, ensure they can't register any others (even with foreign registrars, like with .ru domains), CloudFlare or their reverse proxy would have to abandon them, AND you'd either need to DDOS the site or have its host revoke hosting (and not establish new hosting). Even then, there are options for them to continue.
Daily Stormer was afflicted by, I believe, all three levels; CloudFlare dropped them, their domain got suspended and even the Russians wouldn't give them one, and I believe they lost at least one host. StormFront so far appears to only be affected by the domain suspension.
Of course, losing your domain name is still annoying, the same way getting a new phone number is annoying. But they will presumably just email all of their users using user contact emails and tell them the new domain that way.
