No, it does not, or if I want to be more precise, it's the wrong protection against the threat.

Take a step backwards: what is threat we want to protect from? The threat we're trying to protect from is any person on the public internet who can bypass the normal authentication mechanism. That's what these "script kiddies" are trying to do: access the server using SSH even though they're not authorized. In particular, your superior wishes to prevent these attackers from even being able to start establishing an SSH connection.

What's the standard technology to prevent establishing network connections? A firewall. The standard answer to this problem is to just block port 22 entirely to outside traffic. The bigger problem here is that SSH is available to the public internet at all, and the firewall solves this completely, while the obscure port only hides it slightly. If external access is required, the standard answer to allowing this authorized access is a VPN into the network. This would both block script kiddies and knowledge attackers who could figure out how to find the port you're actually using.

If you lose to 1% of attackers, you still lose. You need protections that work against 100% of attackers. If instead you are successfully blocking 100% of attackers (so far), then you must have more robust protections in place. These other protections make the obscure port irrelevant, and if that is (hopefully) the case, all using the other port does is confuse people who are less familiar with the actual security practices like yourself and, quite probably, waste the time of people trying to gain legitimate access when they try to connect (new system not configured and have to ask someone the correct port, admin forgot to tell person about the nonstandard port and person has to bother them again to diagnose the problem, etc.).

This is why we harp about "security through obscurity" and Kerckhoffs's principle. We should avoid distracting ourselves with practices that don't actually protect us. We should save our time and effort for protections that actually work and avoid the false sense of "security" that these obfuscation methods give us.

No, it does not, or if I want to be more precise, it's the wrong protection against the threat.

Take a step backwards: what is threat we want to protect from? The threat we're trying to protect from is any person on the public internet who can bypass the normal authentication mechanism. That's what these "script kiddies" are trying to do: access the server using SSH even though they're not authorized. In particular, your superior wishes to prevent these attackers from even being able to start establishing an SSH connection.

What's the standard technology to prevent establishing network connections? A firewall. The standard answer to this problem is to just block port 22 entirely to outside traffic. The bigger problem here is that SSH is available to the public internet at all, and the firewall solves this completely, while the obscure port only hides it slightly and doesn't actually prevent the connections. If external access is required, the standard answer to allowing this authorized access is a VPN into the network. This would both block script kiddies and knowledge attackers who could figure out how to find the port you're actually using.

If you lose to 1% of attackers, you still lose. You need protections that work against 100% of attackers. If instead you are successfully blocking 100% of attackers (so far), then you must have more robust protections in place. These other protections make the obscure port irrelevant, and if that is (hopefully) the case, all using the other port does is confuse people who are less familiar with the actual security practices like yourself and, quite probably, waste the time of people trying to gain legitimate access when they try to connect (new system not configured and have to ask someone the correct port, admin forgot to tell person about the nonstandard port and person has to bother them again to diagnose the problem, etc.).

This is why we harp about "security through obscurity" and Kerckhoffs's principle. We should avoid distracting ourselves with practices that don't actually protect us. We should save our time and effort for protections that actually work and avoid the false sense of "security" that these obfuscation methods give us.