Difficult to know for sure what such "fix its" may actually be doing or trying to do.
For the most part they probably check version information for hardware and software. Older or buggy versions may be updated or flagged in some manner.
Checking for valid licensing and and device serial numbers perhaps to screen out pirated software and counterfeit products.
The "fix it" may also restore settings back to some basic configuration and make attempts to communicate to the manufacturer's website or some other test site/device.
That is necessary because end users often create problems when changing settings or making tweaks. Especially if such changes were made via some internet website and not the manufacturer's website.
Some fix-its work very well. Others not so much.
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That all said:
An X-box fix-it should not be making changes to the router.
At most, perhaps the fix-it would reset to some default network configuration within the X-box which may or may not work with the host network.
What router tweaks have you made?
Make and model router?
Is there a specific network problem involved?