What's the state of things in late 2019?
There are currently two ways to install an external GPU to a laptop.
By using an internal PCIe connection and a cable leading to a PCIe GPU enclosure.
This is often done through the port for the WiFi card. If your laptop is new enough it might have a spare M.2 port that supports PCIe for an SSD or a cellular modem.
By using a Thunderbolt 3 port and a cable leading to a PCIe GPU enclosure.
In the near future USB 4.0 will incorporate the Thunderbolt 3 protocol. This will likely cause the current Thunderbolt 3 enclosures to be supported by a large number of laptops with USB 4.0 support.
Does such a thing exist for USB-C as well?
No and yes.
"USB-C" is a specification supporting USB 3.x. There is no support for exposing PCI express through USB-C.
Thunderbolt 3 is an entirely separate specification that supports USB 3.x and PCI express. It also uses the USB-C connector but the port is considered a "Thunderbolt" port, not "USB". Thunderbolt devices usually require a superior quality cable too.
What's the optimal solution for me?
It looks like the adapter you linked might work with the WiFi slot, but it will be significantly slower than a Thunderbolt 3 connection, and may bottleneck the GPU. If your laptop is new enough you might need to use an M.2 version of a kit like that, since many WiFi cards come in an M.2 slot now.