PowerCon and True1 power connectors have become very popular in live event production, and many lighting fixtures now use one or the other. They work well, especially in touring applications where the gear is traveling to other parts of the world where the connectors are different. In the old days we would have to rewire the power cable with a new connector, and now we can simply change the entire cable without any tools. But you're right about these connectors not accepting the thicker type S cable, but they do work with type SJ.
Type S is what's known as "extra hard usage" cable. It's designed to be routed on the floor, and it can be stepped on, run over with a forklift, and generally abused and it will hold up well. Type SJ is "hard usage" cable, and it's not designed to be stepped on or run over by a forklift. It's supposed to be protected from that stuff.
It's true that the National Electrical Code says that, in theatres, audience areas of motion picture and television studios, performance areas, and similar locations, flexible cords and cables, including extensions, should be listed, extra hard usage cords and cables (i.e., type S). However, it also says that listed hard usage cords and cables (type SJ) are allowed for use as power cables on fixtures as long as they are not longer than 2 meters (6.6 feet), it mates to a listed, panel-mount connector on the fixture, it's protected by a circuit breaker or fuse not more than 20 amps, the fixture is a listed device, and the cord is "not subject to physical damage."
It also says you can use type SJ for Socapex breakouts as long as the longest cord in the breakout assembly is not longer than 6 meters (20 feet), they are attached to a pipe, truss, tower, scaffold, or some other structure to protect them from physical damage, and they are protected by fuses or circuit breakers not more than 20 amps. Lastly, type SJ cable can be used for two-fers as long as they are not over 2 meters (6.6 feet) long.
Of course, if you make your own adapters, they will not be listed, so they will not be code compliant. There are manufacturers who make and sell listed adapters, so now that you know the code, you can make an informed decision about what to do. Be safe!