DNS doesn't propagate. There's no reason those name servers would have the name server information for the domain name you supplied unless a DNS client of those name servers asked them to resolve DNS records for the domain... or the DNS propagation checker asked them to query for the name servers of the domain name you supplied.
So it seems to me that what those DNS propagation checkers are actually doing is querying each of the listed name servers for the domain name you supplied, thereby forcing those name servers to query for the name servers for the domain name you supplied, thereby caching that information at those listed name servers.
DNS servers don't cache the whole of the entire DNS namespace nor do they cache all of the name servers for all of the registered domain names. Imagine if they did? What kind of scale would be needed to handle a database of that size and that type of workload, especially considering the near constant change to DNS records for all registered domain names in a single day? My name servers know nothing about your name servers or your DNS records until I ask my name servers to resolve a DNS record for your domain. Your name servers and DNS records don't get "propagated" to my name servers at any point. My DNS client makes a request of my name servers for a DNS record in your domain and my name servers then work to answer that query. Only then will my name servers have any information whatsoever about your name servers and domain name.
Using a DNS propagation checker may seem like a good way to check that your name server information is being found correctly, but propagation isn't what's being checked. The listed name servers are querying for the name server information for the domain name you supply and if the name servers are found then you get the big green check mark. It's a good test to see if your name server information can be found, but calling it a DNS propagation check is propagating (no pun intended) a myth about how DNS works. As you've seen though, those checkers can mask other problems... without giving you any clue as to what those problems may be.