Why are Avid brakes so difficult to bleed?
It's the design that makes them harder to bleed. Although an 'open' hydraulic system Avid's Taperbore resembles a 'closed' system, one that operates in a vacuum. The Avid system uses a small reservoir fixed amount of fluid in the system and a bladder around the master piston with air in it. Shimano and other brands use a larger reservoir with a bladder separating the brake fluid (in the case of Shimano mineral oil) and air. In both cases air is used to compensate for the fluid in the system expanding.

Because the Avid system operates in a vacuum it requires all air to be removed from the system and that is why you need the two syringe set, other designs simply require fluid to be topped up in the reservoir and air worked out. The Avid elixir system is further complicated by a design that traps air within the taper bore reservoir adding an additional level of difficulty when purging air.

Avids also use DOT 4 or 5.1 which is more difficult to work with than other fluids. Different brake fluids have different characteristics (boiling points, freezing points, amount of expansion etc.). Avid specifies DOT 4 and 5.1, which is a pain. The brake fluid is a solvent so you don't want to get it on your hands but it isn't going to kill you. DOT fluid is hygroscopic, it attracts water so although you can buy it in larger quantities cheaper than through Avid it doesn't have a good shelf life once opened and probably shouldn't be opened again.