That’s a good chip. As a rule of thumb, Realtek = best, Asmedia = good, JMicron = garbage. JMicron adapters run super hot and draw a lot of power, leading to low speeds and dropped connections.
The reason I suggested a Live CD with persistence is that they are better at autodetecting stuff on the host machine. You can definitely install an actual system on the SSD but it will make assumptions about things like the GPU for example – won’t expect to have to swap it at boot, you’ll have to do it manually. Or you can run your desktop environment with a pure software driver but that may get a bit annoying at times, depending on what you want to do with it.