Is support for USB Type-C alternate modes (such as MHL and DisplayPort) entirely the responsibility of the USB controller, or is the responsibility shared among other devices? For example, if I wanted to use a USB-C-to-HDMI cable to connect my computer to my monitor, would I just have to get a USB add-on card that supports the MHL alternate mode, or would I need a graphics card that has its own USB-C port? Similarly, if a motherboard has a USB-C port on it that doesn't support any alternate modes today, will any hardware additions or BIOS updates ever make that port support them?
For USB-C alternate modes to work, there has to be a physical connection from the alternate signal source to the corresponding pins of the USB-C port.
Unlike some older USB video adapters that actually include a small external GPU and transmit the display content as USB data, USB-C alt modes really just patch through a physical signal that the USB controller doesn't even understand. The controller is just used for negotiating what that alternate signal will be and which of the 1-4 possible connections will be used.
For discrete graphics cards, it seems unlikely that such a physical DisplayPort (or MHL) path exists from the GPU to the mainboard's USB-C controller.
Even for integrated processor GPUs, there needs to be an additional USB-C controller that negotiates the appropriate alternate mode and physically routes the signals.
The principles are explained in this presentation of the DisplayPort alternate mode.
If your current hardware doesn't support any USB-C alternate modes out of the box (or is at least prepared for it), I wouldn't expect a software update to be able to change that.