Is there any way I can "upgrade" my HP's normal USB-C or USB-A to a Thunderbolt Port?
No, supporting Thunderbolt requires a Thunderbolt controller in the computer. You can't upgrade a port from USB 3.x to Thunderbolt with software or a dongle any more than you can upgrade a USB 2.0 port to USB 3.x with software or a dongle.
If all you want is a dock for two displays, 2 or 3 USB-A devices, and passthrough power to your laptops with USB-C ports then there are plenty of options that should be under $150. The laptops will have to support DisplayPort MST on the USB-C to find a dock in this price range with dual video support, unless you find a real deal on a DisplayLink dock.
I've tried the DisplayLink approach and updated the drivers but it still isn't working.
I'm not sure I understand. DisplayLink is a brand of USB to video hardware and you can't add DisplayLink to a dock with drivers. To get DisplayLink means getting a dock with the DisplayLink hardware in it. DisplayLink docks will require drivers to provide video, DisplayPort docks will not require drivers for video.
Don't get too hung up on "Thunderbolt compatible", I've seen this term abused into meaninglessness. All "Thunderbolt compatible" means is it won't burst into flames if you plug it into a Thunderbolt port. If a dock plugs into a USB-C port then it is by definition "Thunderbolt compatible", unless they really screwed up the wiring. Being "Thunderbolt compatible" is not the same as requiring a Thunderbolt host to work.
Also abused into meaninglessness is "universal docking station". As best I can tell is that this term describes a dock that can connect to laptops that may or may not support DisplayPort on its USB-C port, and may or may not have a USB-C port. They get video by using a DisplayLink chip in the dock. If you have laptops that support DisplayPort MST then using DisplayLink will likely be going backwards in video performance. Most any laptop with DisplayPort on USB-C will do better than DisplayLink can offer and for lower cost.
Thunderbolt docks (not "Thunderbolt compatible" docks) are often quite expensive because they require the bandwidth that only Thunderbolt can provide. A dock to get two video ports and 3 USB-A ports will not require Thunderbolt to work. Ignore the terms "Thunderbolt compatible" and "universal docking station", this is meaningless marketing speak.