Is anything stopping me from getting a 5V 20A power supply and wiring up
20 Micro USB cables in parallel
Not really. If you want something simple, just put a 2-3 A resettable polyfuse on each VBUS wire on your board, and use good grade full-sized micro-B cables. The simplest way to provide your devices with up to 1.5 A, short D+ and D- wires together and leave them floating (in each cable). This connection will inform attached devices that the power source is "Dedicated Charging Port", aka DCP (which is an adoption of Chinese Federal Standard). Most devices understand this "charging signature" and will take about 1 A from the ports (up to 1.5). So you might need a 30-Amp power supply.
If your devices are not using "fast charging", you can leave D+ and D- unconnected, and devices will take no more than 500 mA (if they are good-behaving devices).