I vote for installing a small subpanel, which you could feed from your 60A outlet.
You could probably surface-mount the small panel near the 14-50R receptacle and feed it by hard-wiring a length of 6/3 MC cable into it, clamp that to the box with a jake clamp and fasten it to the wall as many times as code calls for, put a 14-50R plug on the other end, and just plug it in.
Then you could use some 14/2 MC connected to a 15A breaker, or 12/2 MC cable connected to a 20A breaker in the subpanel, to feed a surface-mounted outlet box or two. Make sure the neutral and grounding buses are isolated in the subpanel, that the grounding bus is bonded to the subpanel housing, and that you have 4 wires running all the way back to the main panel (2 hot, neutral, grounding).
EDIT/Correction: Those LED screens are not 60A devices. You'd have to be creative to attach those devices to the 14-50R receptacle, and it isn't legal to put 15A or 20A receptacles on a 60A circuit because the receptacles themselves aren't protected by the circuit breaker. They have to be on branch circuits with their own 15A or 20A circuit breakers.
Hypothetically, let's say you invent a machine that requires 40A or 60A split phase service. The machine has something (or things) with a big draw, like motors and/or heating elements. It also has a number of small-draw devices built in, such as small light bulbs and/or LED lights on a panel, a couple of small LCD or LED readouts, and a 14" color LED screen. That's all good, you can feed all those small devices off of one or the other leg, fed through a 60A breaker. But you're also going to design the machine with built-in circuit breakers or fuses on the internal branch circuits to protect those smaller components.
Those Powerconn connectors are slick, but you're putting them on what amount to extension cords that are hard-wired into a 60A circuit and I'm guessing the cords aren't made with #6 wires? So you still have a loose, undersized extension cord that can serve up to 60A for a short period before it overheats and starts a fire. And that's probably also not legal.
Balancing the branch circuits isn't going to be an issue (unless your neutral comes loose). If your hot wires are connected correctly (connected to different legs in the panel, using a double-pole breaker), and the neutral is good, then the load on the neutral will never be bigger than the capacity of the breaker(s). The neutral (grounded) conductor only carries the imbalance of the loads on the two hot conductors. The neutral can't be overloaded, because the power from each of the hot legs cancels out the power from the other. If one leg is fully loaded and the other has no load, the neutral will only carry as much power as the loaded leg. Add any load to the other leg, and the load on the neutral goes down (not up).