Is there a way to use the i2c interfaces on a standard PC mainboard to control a device?
Yes, this should be possible. The PCs do often have several I2C ports exposed to the outer world:

The DDC pins on HDMI, DVI or VGA are actually using I2C. I have successfully controlled an IO expander chip through this as a PoC a few years ago, also powering it through the power pins there originally meant to power the DDC identification ROM.

The I2C devices mentioning NVidia in the name are likely for DDC on the various video output ports. Finding which bus maps to which output port will likely be a trial-and-error process. Or you can try scanning all available I2C busses for a device on your known address.

You could make a simple breakout board with say a HDMI connector hooking it up to your whatever, and only use the DDC-related pins. It should be rather convenient too, with only a single cable.

So yes, this should be possible, if perhaps a little janky.