Electrical – typical value of contact resistance: 1) banana connector, 2) solder joint

The wire gauge you need is a function of several things.

• The acceptable voltage drop or power loss (that appears to be the only thing considered in the website you linked). The voltage drop (and power loss) is proportional to wire length and inversely proportional to the cross-sectional area of the wire- in other words inversely proportional to the square of the wire diameter (assuming constant current).
• The acceptable temperature rise. This is a function of the number of current-carrying wires bundled together, the environment (maximum ambient temperature and air pressure or altitude, for example), the insulation type, the wire type (some types of wire are plated to withstand higher temperatures than bare copper without corroding).
• Regulatory requirements and other considerations- for example, the wire may be rated for 200°C insulation, but you might not want the wire to run that hot.
• Fusing- the fuse or circuit breaker should protect the wire in the case of faults such as overload or short circuit.

Very short lengths of wire can depend on heat sinking through the ends (indeed, in a vacuum, that may be the main heat loss mechanism), but usually that's not taken into account.

Normally you'd run through a checklist such as the above to make sure ALL the requirements are satisfied simultaneously, so you might find that using PTFE insulated wire allows you to use AWG 18 wire, but because of the voltage drop limitation you'll have to use AWG 12 wire.

Modularity

As @helloworld922 and @Ignacio Vazquez-Abrams point out, a thick wire is extremely difficult to handle in the tight confines of a computer chassis (which are getting ever smaller to meet consumer preferences). The power connector is attached to a card which must exit the bend radius within the specified spacing to the adjacent card so there is limited ability to just make the wire thicker. Further, accommodating thicker wires presents fewer options for placement and routing of the PCB making design of the PCB and guaranteeing sufficient airflow around the cable and connector system more difficult.

By using multiple connection points, the spec can accommodate future reductions in power demand or differences between high-end cards with very large requirements and smaller/low-end cards with less. This is especially important in the modern computing era where performance comes from parallelization (more cores, rather than higher performing ones).

Consumer Safety/Confusion

@Tom Carpenter writes, "That and if they used the same connector but with thicker wires, someone would inevitably have an old model supply with thinner wires and use that for a new card without stopping to check. An important lesson in engineering is: Assume the user is an idiot. Many won't be, but there is always one!" -- It's apt advice. Further, many users would be confused by the subtle distinctions.

Cost

Cost is not obvious, but it is likely much cheaper to implement it the way it has been done in the PCIe spec. The majority of the cost in short-run cable assemblies is actually the manufacture of the terminations (end connectors) and the assembly of these terminations to the wire -- not the wire itself. Thicker cables have much lower yields when using crimp or insulation displacement manufacturing techniques and the larger cables necessary to accommodate the PCIe power goals would likely not fit in the existing tooling and machines used to assemble the molex, audio, and other PC wiring. PC cabling is manufactured in such large volumes that even small differences in yield make a substantial difference.