The server "going down" is one of many possible issues. You need to take into account the the "server" is a collection (in this scale) of many distributed copies of a number of "servers". And I put it in quotes because there is a vast difference between a physical server (the computer) and what we mean by server, which is server, the program that provides a service.
A site like chess.com needs a great number of physical machines running "copies" of those services and databases, with an number of inner machines that update the final status of the site globally (ie, for all machines running those core services). That is the site core.
The "decision": "which machine (or, most likely, set of machines) will provide service Y to user X" is made by using a number of intermediary services on different machines, colloquially known as load balancing/load balancers.
At (the very) least for those that are somewhat more adept at such concerns, a set of basic information (regarding one's location and connection, of course), specifically about the state of performance (basic latency, load time server-side, state of mirrors etc.) must be provided along with an estimate from the site itself (eg. "there is very high usage of the database servers, games are dropped" or "CDNs in <user_country> are experiencing technical issues" etc.).
Hello! As a fellow server admin (just an enthusiast, not chess.com level, but with about 16 years of experience) I am proposing that a general, not detailed, server status tab be added, either as an optional "gadget" or as a mandatory footnote-style section, because when the server(s, CDNs?) suffer an issue, many games are lost or cancelled. I lost a number of games in the last half-hour due to spontaneous cancellation, that was caused either by maintenance work or by some infrastructural issue. Players should know when they should avoid playing, because time and points are lost when one does not know if they can play or not.