IC's are not in yet.I don't see any ICs in the photo. But if you are relying on the internal impedance of the 4001 gate to limit the LED current, the gate will not last long. You really do need current limiting resistors (1 per segment) to prevent driver and display damage.
An LED is like a signal or rectifier diode in that it has no mechanism for limiting the power it dissipates. An LED rating of 4 V and 20 mA means that if you externaly limit the current through the LED to 20 mA, the voltage drop across it will be 4 V. If you apply a 4 V source with lotsa current available, the LED will try to pass all of it and fail.
With a 5 V source, you need to drop 1 V at 20 mA across the current limiting resistor. With Ohm's Law, this works out to 50 ohms. 47 or ohms are the closest standard values at 5% tolerance.
But lets look at fwd V vs current. My bench PS is adj 30v and will handle 30v@10A. I put 4v on LED segment and get 20mA of current. So what exactly do you mean the LED will pass all the current? LED's have fwd and reverse bias impedance.
With just the diodes I am driving all 7 segments at just 60-70mA, around 10mA each. The LED's did not create a short from anode to cathode, if they did then I would have seen my PS go to max amps for a brief sec and then LED fizzle to its death.
So please explain for me, what am I missing?