I think the part that is intimidating to me is the power/clock/microcontroller integration. Maybe the gateway is to design a PCB that just connects the LEDs and over time add more and more components to it.

I bit this particular bullet a couple of months ago. Turns out that a very cheap board like an Arduino Nano Every has a power voltage regulator (that will accept 6-21V supply) and a built-in clock, so you don't need to integrate anything.

(As it turned out, I wanted a more stable clock signal for timing purposes than the arduino clock, so I attached a 32.768kHz crystal to one of the inputs, and that also was very easy. I also attached a 4-digit 7-segment display, as there are enough I/O pins, a thermistor, several buttons/switches, and a stepper motor controller. It was easier than I thought. Give it a try.)

If you’re just starting then imo it’s easier to go the other way. Start with a fully-built dev board like a Wemos D1 mini. Keep building things and as you run into roadblocks and learn how to get around them you’ll naturally develop the knowledge to get further into fabbing your own integrated PCB.

You might look up various DIY arduino designs (starting on a breadboard). You can start with very barebones selection of components, move it to a proto-board, and then modify it to your needs. Adafruit and Sparkfun have great resources for how everything they sell works.