Lots of things to do!
When refereeing CS:GO, the usual process is as follows:
Check for match start: if match is starting normally (ie automated system is working fine) we let the match start on its own, if there's a holdup, we have to check.
There are several things which could cause a holdup: the mapvote may not be completed, in which case I'd prompt the teams to finish it. Then, all players need to login to the AC, and make sure it's running properly. A common mistake is for players to start the game before activating the AC. Now, sometimes the AC has issues, so I need to sort these out too. Then, I need to make sure all the players are in the server: sometimes they'll get kicked out for no apparent reason: I can see a detailed error message from a browser API/GI to find the issue. An example of the issue would be that a player isn't registered with the team they wish to compete with. In that case, I take their steam ID and their ESL ID to input it into our system, then ask them to restart their game and AC. If a team can't field a valid team of players I give a defwin to their opponents.
Then, the server may also have issues: in this case I check it out personally to see if the server sucks. If it does, I change the server, and if an entire server group is affected, I change the server to a neutral location (so no team is given an unfair advantage). Then, sometimes the players experience packet loss from the server. In that case, I change the server location, which changes the packet routing. If the issue persists, that's explained in one of my recent threads, called 'Packet loss on ESL servers?', created following the NiP-Anonymo drama. Once that's all sorted, the players can start the match. Once started, the match is pretty much automatic: the players can choose to pause etc, and if need be I can watch the live chatlog and serverlog from the API/GI. If there are insults, either to me or any other involved party, I open a ticket with the evidence, and send it to the relevant admin squad, who will penalise the offender accordingly. The same happens with a cheating accusation (I'm part of the anticheat squad, but not the insult squad). Although there are other problems to solve, this is the process for 95% of the matches at grassroots levels. Casters can request a GOTV link, which goes through my approval.
Pro play is slightly different, because production may want to do extra stuff, which we accomodate! This also includes monitoring of comms and cams (of the players, not their playing PoV: that's handled by observers).