How to control a lightbar with 20 individual modules?
For the electronics part, the best method I found was using 20 MOSFETs, 


Yes, best solution...


  because they can be switched quick enough


...but wrong reason. 

Good reasons to use FETs are: easy to drive, cheap, low RdsON so they don't heat up, you get two in a convenient SO-8 package...

Blinking a light at 10 Hz is really fast visually, but it's really slow from an electronics point. FETs can switch in tens of nanoseconds, which is fast, and completely irrelevant here. If you want to blink a light then you want your FET to switch slowly like 1µs in order to avoid EMI. This is very simple to do, just put a resistor on the gate, a few kOhms.


  designing and printing a PCB seems overly complex, but if it can be made simple, saving a few hours of soldering is always a good idea :)


Either spend a few hours designing a pcb and have it printed by a shop like pcbway, or breadboard it... best option will depend on the controller chip which will be difficult to breadboard. Also on a PCB you can use SO8 dual FETs.


  For the controller, my initial thought was using an Arduino Mega, which has plenty of I/O, but I had reliability problems with using it in industrial environments


Arduino is not intended for industrial environments.

Picking a microcontroller is a bit of a touchy-touchy topic which can devolve into holy wars. Let's be practical: if you know one already, pick this one.


  other than that I couldn't find anything commercially available with this many pins. 


Personally I'd pick an I2C IO expander with 8 outputs, and make a board with it and the corresponding number of MOSFETs. I2C is a bus and chips have addresses, so if you make 3 identical boards you get 24 outputs. You can also make a 20-output board, but it is less reusable.

Note this assumes your LEDs are wired with resistors and expect to be driven with 12V. If your LEDs require constant current drivers, then you will need to provide them.

In fact you should probably be able to find such a ready made board online. I made a quick search and didn't find one, though.

Here is another example ; this is a I2C controlled chip with 16 PWM outputs. If you wire this to 16 FETs then you got 16 dimmable outputs.


  I also thought using shift registers to work around the pin number problem, but I'm worried it might be too slow for rapid flashing.


This is also an option. Don't worry about the speed, these are about 1000x faster than you need.

In fact your main problem is you got too many options...