The fundamental problem is not Windows or Linux or any other software, the very basis of this mess are the people: Developers are dumb and lazy, their bosses are dumb and greedy and the admins are constantly overworked, because they have to fix the bullshit, that the first two groups are throwing at their doorsteps. Agile development only added to this cesspool, as everybody only tests for the predefined outcome of his little change and no one views the big picture, where interactions and synergies come down like a sledgehammer... after over 25 years as an admin and part-time developer, I left this industry to keep my sanity!

Don't get me wrong: I'm not "pro-Windows" or "pro-Linux" in any way. They both are valid options, but you have to consider the consequences. Keeping a Linux system up-to-date is a bit more work than a Windows system, but the Linux ecosystem is a bit more "snappier" about security issues...

We should agree on a simple fact: running a webserver on a 14yr old system just isn't a good idea!

Depending on your scope of responsibility and the number of customer systems, licensing costs are your least concerns. Right now, I'm in charge of ~150 Linux servers and ~40 Windows servers. The customers on Windows had no problems with 30 minutes of downtime on a Wednesday night (they know Windows update!), but the Linux customers were bitching around like puberting teenagers...

You never had to deal with C-level, right?

It has to be really smart to appear dumber... :)

This answer is as dumb, as it can get by biological means; not sure, if a specialized AI could beat that...

It's not about Windows vs. Linux, it's about management vs. IT. A 14 year old Linux server will not fare any better, in fact, it's a lot easier to build a shitty server with Linux than it is with Windows. I've done web servers with VMS, OS/2, Unix, Windows and Linux and you can build decent servers with any system, as long as you know, what you're doing. The key has always been to convince the CFO, that it's worth it to keep the system up-to-date...