Thunderbolt cable that only works as USB 3.1

Are there ways to test if the cable that I have is a working Thunderbolt 3 cable?

Yes. Purchase a cable from a known vendor with a reputation for quality. I tend to stick with Anker, Cables2Go (C2G), Belkin, and Pluggable (cables, not peripherals) and even Apple (well, whoever Apple selected to OEM for them).

C|Net summarized it perfectly in their article Here's everything you need to know about USB-C and Thunderbolt 3:

• All Thunderbolt 3 cables will work as USB-C cables.
• All USB-C cables will work as Thunderbolt 3 cables as long as they are good quality cables.

^{Emphasis mine}

It’s when you start rummaging around in the bargain bin looking for low cost USB and Thunderbolt cables that you end up with cables that just don’t work the way they’re supposed to. Spend the few extra dollars up front and it’s something you won’t have to worry about (meaning spending hours diagnosing and days waiting for your delivery from <insert favorite online retailer here>)

If you plug a TB cable into a TB3 port and hook it up to a USB device, it’s going to show in the “USB section.” Likewise, plug it into the same port but now the device is a TB device (like a dock), ot will show up in the “Thunderbolt section.” In other words, the cable doesn’t define how the device appears -the device snd the port do. It’s also important to remember that Thunderbolt 3 carries USB 3.1 but not the other way around.

As for your LG cable, I wouldn’t preclude it being of suspect quality. Just physically comparing the Apple cable (which I already owned) and the LG cabled included in the box, the Apple cable was clearly of higher quality. I’ve chucked more “included” cables because of horrible build quality than I care to remember. Remember, a lot of this is to meet a requirement, check off a box on a list somewhere and most folks won’t notice the difference

I'll step through some obvious reasons this might not work, so don't feel offended if this sounds "basic" and you've done this already. The setup you describe should work so it comes down to a bad setup or bad hardware.

The first thing is to see if the cables are in fact Thunderbolt cables. They should have USB-C connectors on each end and the Thunderbolt icon on one or both connectors. If you don't know what the icons look like for the different connectors/ports then take a look at Apple's support page on this: https://support.apple.com/en-us/HT201736

The second thing to check is that the devices needing the most bandwidth are connected closest to the computer. Connect the Thunderbolt cable to the Thunderbolt ports on the Mac and the display. Any USB-C devices connected in this chain should be connected to the ports on the back of the display. It's best to connect any USB-C devices directly to the Mac if you can since they can "steal" needed bandwidth from the display.

A Thunderbolt cable is "dumb", as in these cables don't often have any electronics in them to identify them as a Thunderbolt cable when connected to a Mac. A Thunderbolt cable when used to connect a USB 3.x device will look like a USB cable to the computer, it doesn't know the difference.

If walking through the wiring and checking that it's like I describe then perhaps something is broken. Do what you can to verify all cables are working, as well as the display and devices.