I would use something cheap and simple, like an Arduino, and an "LED display driver" chip. Then it does not matter how many PWM signals are on the MCU. In fact you could use a tiny MCU, maybe even an 8pin device. The "LED display driver" handles all of that.
Typically they drive LEDs directly, so dispense with any need for external transistors. Many have constant current source drivers, and so dispense with any need for external current-limiting resistors for the LEDs.
Many manufacturers make such chips, for example TI display drivers
Also Maxim Integrated
Many do PWM themselves. The PWM data is loaded over a serial channel (which could be done using SPI or bit banging), and the chip 'just does it'. Many use one external resistor to set the overall current for all of the LEDs.
Some come in DIP packages that you could prototype on a breadboard.
Some cost under $2.
They can also be 'stacked' end-to-end, so if you need to add more PWM outputs, you'd just add another part.
If you wanted to strip out the MCU completely, you could get a USB-to-SPI chip. I think it is more awkward than using an MCU, e.g. an Arduino, but YMMV.
[General remark: Treat hobby-grade sources like SparkFun with a pinch of salt. Their objective is to make things easy, fun, exciting, practical without getting bored with integrals and derivatives. As a result, you get loose statements occasionally.]
In some books, PWM is called a pseudo-analog signal. It has only high and low voltage level, but timing is continuous. Timing is either truly continuous, or it has high enough digital resolution where it can be considered continuous for practical intents and purposes. When PWM is converted to an actual analog signal, that's always done with a low-pass filter of some sort. Here's are some examples:
- So called "poor man's DAC", which consists of a PWM output from a microcntroller followed by a low-pass filter.
- Buck converter
- Class D amplifier
Try the LTC6992 from Linear Technology for size: -