Man, I just do not get what's happening here. It sounds like there are just SO MANY things WRONG about how people could let problems like this affect them.
First: it's harming terminal servers... What? Why would you need to use a browser on a terminal server? I just don't get that, first and foremost. By and large, a GUI should be a luxury, if you're remoting into a server. Even for loopback purposes, a browser should only be needed for a web server.
Second: Why is Chrome, but most importantly GOOGLE Chrome left as the only option available to users? Obviously, it's been made clear in recent news that Microsoft has adopted Chrome as its browser core, but the Microsoft distribution should be on a different release channel, no?
Third: Why is Google not firewalled and blacklisted, such that Chrome can't phone home. Seems like an obvious disease vector. We all know that Chrome routes around policy controls, by installing to user home directories, so that it can thwart frisky, tyrannical administrators that would seek to prevent users from installing wonky appz, progz and warez.
Fourth: Why no enterprise LTS install. Google used to supply this to Windows Enterprise deployments. And if they did, it's, by definition, supposed to be a slower, more stable release channel (long term, often meaning six months, or perhaps years, with only incremental urgent security patches, for emergency zero day coverage). Shame on Google for crossing that line, but shame on enterprise subscribers if they installed non-professional consumer software on not just enterprise machines, but servers no less.
Fifth: If you need browser on your terminal server, in order to get things done, you're doing something wrong. But really, if that browser is mission critical to performing actions on said server, and you are dependent on it, then I question your skill level, talent and wit in general. Browsers just aren't that special. There's no way Chrome should be such a crucial component, while dialed in to a server, such that only Chrome can accomplish things that cannot be performed by some other means. Please check yourself, because it sounds like you made your own bed, and you should be made to lie in it, because you are bad at your job. (or dump the whole situation, quit, find some other place to work, if you are subject to the bad decisions made by someone else, because this sounds stupid)
Last: I bring this up, because I can't imagine the Chrome dependency being rooted in anything else. Intranets and JavaScript.
Chrome's singular claim to fame is essentially it being a consistent JavaScript wrapper for client-side UI delivery. If you need to look at page elements that only work via JavaScript, again, you've created your own problem. And if JS is needed on a terminal server because intranet? Holy shit! You did it to yourself, and there's no two ways about it.
You need to correct your situation, and extricate any such workflows that happen to (A.) rely on JS (B.) in a browser (C.) a browser you aren't really trying to control (D.) while remoting into another machine (E.) one that is indeed a server no less.
Listen, remote desktop, firewalls, VPN access, and web technology are all great, but this is no way to run a business. But most of all, this is no such way to run a server.
No company should be able to pull the rug out from under you by making uncontrolled contact with your server, in ANY way. Not because they aren't supposed to. Not because of some imaginary honor system that might forbid such a thing. But really, because it is impossible for the network traffic to even occur.
Google should be unreachable, even if you've chosen to use their software.
If Google is able to push changes to your shared access machine, you're the one in the wrong.