USB 3.1 Gen 2 (SuperSpeed+, 10 Gbps) was designed to work over both existing USB 3.0 cables (the ones with the 5 extra contacts), as well as USB Type C cables.
Since existing USB 3.0 cables (the ones with Type A and B connectors, as well as the micro A and B variants) only contain one super-speed pair-of-pairs (Tx pair and Rx pair), USB 3.1 Gen 2 could only use that one pair-of-pairs and still work over existing USB 3.0 cables. So even when you run USB 3.1 Gen 2 over a cable with Type C connectors, it only uses the one super-speed pair-of-pairs. This also makes it possible to have USB 3.0/3.1-capable cables with a Type C connector on one end, and the earlier USB 3.0-style Type A, B, micro A, or micro B connectors on the other end.
Now you might ask a follow-up question, "Why didn't the USB Implementers Forum (USB-IF, the USB standards consortium) define an even-faster-than-10Gbps flavor of the USB protocol, that uses both super-speed pairs-of-pairs in the Type C connector?" That's a valid question, but I'm unwilling to speculate. It would certainly have been a bigger departure from previous USB PHY designs, in that it would have two separate send and receive data streams that would have to be coordinated. In effect, it would be a kind of parallel interface whereas USB has traditionally been nominally serial.
The way you asked your question exposed a few potential misunderstandings that I'd like to address here:
I know that it is possible to run Thunderbolt 3 over any high-quality USB-C cable
That's not quite true. There are many high quality USB Type C compliant cables that are not suitable for Thunderbolt 3. Thunderbolt 3 is limited to ≤ 0.5m cable lengths if you have a passive cable. To go longer than that (like 2m), you need a more expensive active cable (a cable with special IC chips in it to assist in signal handling).
Why doesn't standard USB-C use both of the pairs
USB-C is not a protocol. USB Type C is the name of a connector and cabling specification; it's not the name of the protocols that are used over those connectors and cables. When doing USB protocols over Type C cables, you're doing USB 3.1 Gen 2 (10Gbps "SuperSpeed+"), or earlier flavors of USB.
The manual of the DELL S2718D says on p. 9
1x USB Type C (Alternate mode with DP1.2, USB
3.1 upstream port, Power Delivery PD up to 45W)
So it's not only a USB-C type port, it also needs alternate mode, so the USB host adapter in the computer you plug it into must be capable of this mode (and not all host adapters are).
Now it might be possible that the USB controller in the monitor is really really smart, and also supports an USB 2.0 connection, without power delivery or display port signals, where it just acts as a simple USB hub. In that case, an USB 2.0 to USB-C adapter will work.
But there's nothing in the manual that indicates it will do that, so the only way to find out is to either ask Dell, or try it out.
Edit
The manual of the LG 27UD88 says on p. 17:
The USB port on the product functions as a USB hub.
Please connect the USB C-C or USB C-A cable to the PC.
So yes, for data only, it can use USB 2.0. But to display an image, you'll also need a PC capable of USB C alternate mode (even though this isn't explicitely mentioned in the LG manual, but that's the only way DisplayPort can be transported via USB C).
Best Answer
The MSI mainboard and the Razer laptop are both USB HOSTS. The connection between two USB link "partners" can be only if one is HOST, and the other is DEVICE.
The Type-C USB specifications do define so-called DRD - Dual Role Data ports, which can turn itself into being a host or device. However, to have such dual-role functionality from a PC, the system must have TWO USB controllers inside, one acting as xHCI-compliant HOST, and another IP (piece of silicon) that can act as USB device (with all necessary attributes as USB class etc.) The external Type-C port then is internally multiplexed into xHCI or DCI hardware block depending on cable identification rules. The mainstream desktop systems do not have this functionality designed in, but all Intel systems that targeted "mobile space" (Atom line of SoC) usually do have this dual-role functionality.
So the answer to the first question is: yes, you can try to connect both systems with C-C cable, but chances of data transfer are slim, unless the one of systems clearly identifies itself as Dual-Role system (aka "OTG") in its datasheet.
Even if a silicon hardware does have this internal USB capability, the system must provide proper handling of Type-C CC (communication) channel, to determine the role of ports, and engage proper software stack. The Type-C port must be designed in a way that no VBUS power is applied until a cable is plugged in, and both ports sort their identities out. Host ports have CC pull-ups, while device ports have 5.1k pull-down on their CC pin. If both ports have the same "gender role", VBUS power shouldn't be turned on. So the answer to the second question is - if both systems are USB-IF certified, there should be no concern of VBUS power collision.
The answer to the third question is negative, there is no need in any "crossover cable", all crossover is handled in muxes behind the Type-C connector. If both ports are DRP, they will periodically advertise their identity, and the result will be selected at random