The reason you get video from the computer with USB-C and not from the one with USB-A is because USB-C has more wires to carry a video signal. Unless the dock is specifically designed to support video from USB it will not provide a video output. There is a popular series of USB to video adapters made by DisplayLink, if you want a dock that provides video output from USB-A then look for DisplayLink in the product description.
That USB-A to USB-C adapter violates the USB spec is a fire hazard, THROW IT AWAY!
USB-A was not designed to handle the power draw of USB-C, adapting a USB-C device to fit a USB-A port can mean drawing too much power from the host computer and causing the port to overheat. There is certainly some kind of over current protection in the computer to prevent damage but this should not be relied upon because it can still mean your dock will not perform as it should.
The people that created the USB specification created it in a way to prevent the possibility of cross connecting two power supplies. The USB-A to USB-C adapter you have violates this protection and can damage the host computers. Again, the USB host computers will have some built-in protection against this. Even so the fuses may not act fast enough to prevent damage to the computer.
The reason the USB-C hub does not work on the USB-A computer is because USB-A was not designed to carry video in a way that the hub supports. There is no driver or adapter that will rectify this. The adapter you bought to connect USB-C devices to a USB-A port is a violation of the USB specification, is a potential fire hazard, will not fix the problem you have, and should be disposed of as quickly as possible.
The only products I am aware of that will provide video from USB-A are those from DisplayLink or those using DisplayLink chips in them. Because USB-A is capable of only 1/4 the bandwidth of USB-C the performance of DisplayLink video is not going to be the same as what can be had from USB-C or from an HDMI port on the computer.
Most importantly, get rid of that USB-C to USB-A adapter before you inflict permanent damage to your hardware. Adapting a USB-A device to plug into a USB-C port is fine because the spec was designed to support that. USB-C is a superset of USB-A, anything USB-A can do USB-C can do. If a device has a captive USB-C cable then it is because it needs what only USB-C can provide, adapting it to fit physically into a USB-A port does make all the connections needed for the device to function. USB-C was made to support up to 100 watts, USB-A was made to provide 12 watts, maybe 15 watts.
Think of this like trying to cook a turkey in an oven that is plugged into an outlet made to run a blender. Your turkey will not get cooked and your house wiring could be damaged.
