Can I somehow use one PWM output to set various LEDs to different brightness?
This is not exactly what you asked for, but if you want to control multiple LEDs with a single signal, I would look into WS2812 (and similar) family of LEDs and LED drivers. They only need one wire signal(plus supply rails) to control a very large number of LEDs (the more LEDs, lower the "framerate" you get).

Any "diy" implementation of a single PWM to drive multiple LEDs with independent brightnesses would require much more than a "transistor array" and capacitors. (although, yes, an IC is an array of transistors, but what I mean is that doing it from scratch might not be viable solution).

If you are willing to use more than one signal, you can look into ShiftPWM (based on 595 IC's), or into i2c based port expanders, such as PCA9685.

TLC5940 might also be useful.