MAX7219 sounds the way to go. Controls 64 basic LEDs each, wired in a matrix as above, so four of them for 200 LEDs. Each LED will be either on or off at any time but you code for the flickering and such.
With the MAX7219 you do not need resistors for each LED, just one resistor per MAX7219. But in fact, the simplest way to use is it to buy a number of the MAX7219 matrix module kits, unassembled like these ones
The point is that you do not install the matrix arrays from the kits themselves - or their socket pins, but just solder to the positions on the PCB and you have a durable and reliable assembly to drive your own LEDs in matrix arrangements.