You have a few options here. Some of these probably seem drastic, but you should consider what this extra work is doing to your free time. Presumably, this isn't a paid position, thus you are spending your free time being stressed because others aren't handling work you believe they should be handling.
Step back and let someone else handle the issues
It sounds like your server management team is fairly small, perhaps as small as three people (including the problem manager). If the server owner is dedicated to at least keeping the server functional, let them handle the changed configuration files. If it is taking you a considerable amount of time to fix something, I assume it's going to take roughly the same amount of time for someone else to do it too. Let them deal with the issue directly, instead of second-hand through your complaint after the fact.
This route does hurt the player base because they end up with a server that is non-functional for a while. However, this would be the same situation if you were to go on vacation and not have access to fix the problem anyway. It's also a good way to look like a jerk if you mention you are not helping right now because the other guy broke stuff again. I propose that you send a message to the server owner ahead of time mentioning that you've spend significant time fixing these problems in the past and due to the increased time commitment dealing with these problems and not working on growing the server, you're going to step back just a bit from the sysadmin aspect. Perhaps mention that, instead, you'd like to spend more time on the community growth side and will be focusing on that.
Leave the community
You mention this is a small and new server. With that, I am assuming you have a player base that either knows the server administration team or knows one of the first few players on the server. You don't have a large player base, correct? My question to you then: Is this really the type of community you want to start? If the server founders aren't interested in maintaining a stable environment for the players, I suspect they are not that invested in the community. This sounds like a couple friends started a server to play together and eventually recruited a few more people to join and play too. Maybe they started accepting "donations"/"payments" and now the server owner makes a few dollars a month off the player base.
Does the player base see or feel a lack of engagement from the administration team?
Backups
Maybe the above are drastic steps and you really enjoy the player base and are slightly annoyed by the other administrators. In that case, you need to work on ways to reduce the time you spend "fixing" problems. The easiest way to do that is backups on the configuration files. This way, when something breaks, you simply put the other file back in place and everyone is happy it's working again.
Backups can be done in a variety of ways. I'll leave it to you to investigate the best way to backup a Minecraft server.
Propose specific changes (possibly with feedback from the community)
What do you want the owner to do to resolve the issue? Are you expecting him to releave the manager of their position? Are you hoping for another manager to help pick up the slack? Do you want the owner to help pick up the slack? Determine what you want and ask for that specific change. Remember that certain requests are going to receive more push back (remove a friend vs. add a new person to help). You need to approach this as a request for help and way to help the community.
If the community is seeing problems, maybe you can do this type of this more publicly too. Without calling anyone out (because that just raises tensions), ask the community for feedback on how they think the community is doing. Maybe they aren't seeing any issues and think everything is going well (in which case, you are doing a good job, despite the tough situation you are in). Maybe they are seeing a different set of issues than you are. Maybe they have ideas on how to make the server better too. Engage the community.
