He's not wrong, it's just that your browser does that automatically every time it loads a web page. That's an insecure-by-default security model and it's why people run JavaScript blockers like NoScript.
A server, however, is unlikely to do this, so the risk profile of something like Meltdown is much lower. It's not nil because if someone does get on your box, whether they're supposed to be there or not, Meltdown can be leveraged to read the memory anywhere on the system, effectively neutering the protections of a multi-user operating system. This is a major information risk by itself since it could leak important things like encryption keys or user login info, but it could also be used to make local privilege escalation exploits simpler. (That said, if a remote user is able to get a shell on your box and submit code to the CPU, it's probably not likely that you won't already have a latent local escalation vulnerability that can be chained to get root.)
So he's right that people who don't execute arbitrary code on their CPU are much less vulnerable, but they're not totally invulnerable because you can't really guarantee that no one will be able to submit instructions to your CPU due to RCEs and the like. Also, you essentially need to trust everyone who has shell access to your server with anything that may be held in the server's memory, including keys, passwords, etc.
People who are considering disabling KPTI will have to decide whether they want to make the extra local attack surface available in exchange for the performance gain under normal operation.