There is an HDMI alternate mode for USB-C but this does not allow for simultaneous use of USB devices at speeds greater than that defined by USB 2.0. The DisplayPort alternate mode provides for simultaneous use of video and USB devices at speeds exceeding that of USB 2.0, this is done by allocating 2 of the 4 "super speed" lanes to DisplayPort and the other 2 of 4 "super speed" lanes to USB. If a USB-C dock has HDMI and USB 3.0 ports then translation from DisplayPort to HDMI is likely done by circuitry in the USB dock, this appears to be the most common case.
There are some USB-C docks that have a graphics processor unit, or GPU, in them to allow computers that do not support the DisplayPort alternate mode to output video to HDMI displays. These docks with a GPU will need drivers to provide video, docks that use DisplayPort will not require drivers for the dock to provide video. The distinction will be made in the documentation for the dock, and is readily apparent with mention of a popular GPU called DisplyLink. That is an unfortunate name since DisplayLink and DisplayPort are so similar. DisplayLink is a kind of USB GPU, DisplayPort is an alternate mode for USB-C that allows for simultaneous connection of video displays and USB 3.0 devices.
There's the MHL alternate mode for USB-C that provides for simultaneous use of a display and data throughput that exceeds what USB 2.0 provides but the MHL alternate mode appears to be rare.
USB4 and Thunderbolt 3 allow for USB data and video to share the USB-C "super speed" lanes by putting the data in packets that is split among all 4 lanes and have the dock reassemble the packets into a stream for the video and USB ports. This can be confusing since a USB-C dock can support one, the other, both, or neither. By supporting neither the above applies where the DisplayPort alternate mode has 2 lanes for video and 2 lanes for USB data. If both the dock and host support the same protocols then they can go into a mode that will split the total 40 Gbps data more efficiently. There's no more total bandwidth than USB 3.1 allows for but with Thunderbolt or USB4 there can be 30 Gbps for one and still 10 Gbps remaining for the other. With USB 3.1 if the video takes 20.01 Gbps then the dock will set aside all 4 lanes for video and leave USB at speeds no better than what USB 2.0 provides.
Using USB4 or Thunderbolt the total bandwidth is divided by time, it sends video data then USB data alternately as called for with video taking priority to avoid having the video drop out or get chopped up. By using USB 3.1 or 3.0 the total bandwidth is divided by space, the lane is dedicated to video or USB with video again taking priority on which gets those lanes. USB 3.1 and 3.0 uses the lanes in pairs, one for data in each direction, so if the video needs more than 2 lanes can provide in bandwidth then there's no space for USB to fit on the cable.