Several years ago, I designed LED signs for a company. We started out using driver ICs from Allegro. When these became long lead time items, we switched to TI parts, the TPIC595. These are a serial shift register followed by open drain outputs. I believe ST Micro also makes a pin compatible version. The MOSFETs in the ST parts cannot sink as much current though.
The SCR idea has some challenges. First, you have to sink to to power line frequency or the brightness will vary. Second the AC input must have a known and controlled peak voltage or the brightness will vary. Third, for very bright displays, we eliminated the current limiting series resistors for the LEDs and just drove them directly. We would typically drive 5V across some LEDs. To do this we had to limit the duty cycle or the LED would burn out. (Within reasonable limits, the important parameter is the average LED current, not the peak current.) The SCR method seems to imply the duty cycle would be either 0% or 50%.