Why would a switch to Linux imply that you weren't concerned about safety, reliability, performance, etc?
Because it's usually motivated by an attempt to save money, since many flavors of Linux are free. This desire to save money betrays an attitude that is more interested in money than in safety, reliability, etc. If you adopt free software in place of payware, you have to assume many of the responsibilities normally taken by the vendor of commercial software. But organizations that switch to Linux to save money typically aren't willing to assume those responsibilities, and so things go wrong.
The total burden of work and responsibility is going to be roughly the same no matter which operating system you use. Many organizations, when they try to switch to free software, are naïvely trying to get something for nothing.
If they simply wanted UNIX, then the obvious choice would be some sort of commercially-supported version of UNIX, which would bring all the technical support and responsibility of a paid vendor with it. The fact that Linux was chosen instead strongly implies that the only motivation was lowering costs, with all other considerations taking a back seat. The catch is that you cannot lower costs that way, all you're really doing is shifting them around (instead of paying money to a third party, you'll be spending it on payroll for your own employees).
I wouldn't run any important or safety-critical server software on Windows.
Neither Windows nor Linux is appropriate for mission-critical or safety-of-life software. For that you either need a mainframe (in the case of business software) or an embedded system (for safety-of-life software). You can use Linux or Windows for the latter, but not just off the shelf. And for all potential Linux applications, I prefer UNIX or a UNIX descendant instead.
On the desktop, only Windows or (in some cases) Mac OSX is appropriate, unless the desktop role is very tightly and deliberately constrained. For servers, in most cases, I'd install UNIX or its immediate relatives. Linux is popular mainly for reasons that are unrelated to technical considerations. I wouldn't put Windows on a server unless it had to support something that runs specifically on Windows, such as Microsoft Exchange Server or Windows domain management.