Probably for the same reason Apple and Google both produce their own browser - it's too important a part of the user experience to delegate responsibility for it to a third party.

It's unlikely Google would suddenly turn around and say they're no longer producing Chrome for Windows, for example, but it's not impossible.

Also, IE, or rather Trident, is a core OS component. Applications rely on it for displaying web pages and HTML. Applications can also be written in HTML and JS, both HTML Applications (since... Windows 98, I think?) and WinRT applications (since Windows 8).

I never understood why they simply could not use Trident for OS calls and let IE have its own engine. It seems like such a straightforward solution.

It is like building a computer and soldering in the GPU, just because you want to reference the PCI port statically.

The main IE components has been in System32 since IE3 I think.

Yeah you're probably right. They want the experience to be cohesive with the rest of Windows. Last time I looked Chrome still doesn't have a metro mode besides just opening up in full screen mode, so there's one reason to keep IE around