Make seasoning blend: Combine all ingredients in a small bowl.
Prepare meat: Working over a large bowl, sprinkle a piece of meat on both sides with seasoning blend. Set in bowl and repeat, layering seasoned meat. Add beer and mix with hands until evenly moistened. Let stand while you prepare the fire.
For charcoal, ignite a very full chimney of half mesquite lump charcoal and half mesquite charcoal briquets on firegrate of a charcoal grill. When lit, 20 to 25 minutes, dump coals onto firegrate and add 1/2 chimney more lump charcoal; let burn 5 minutes to ignite. Arrange coals in a slope from one side of firegrate to the other. Set cooking grate in place. For gas, heat grill to very high (550° to 650°), then lower heat for one burner to medium-high.
If needed, cut meat in pieces to fit grill. A portion at a time, start grilling cactus and onions over cooler part of fire, covered and turning occasionally, until cactus is olive green and onions are just tender (4 to 8 minutes total, up to 15 minutes for spring onions). If needed, drape onion tops over cactus to keep from blackening before bulbs are done. Transfer to a board. Cook meat a portion at a time over hottest part of fire, turning once, until medium-rare, 1 to 4 minutes total (3 to 8 minutes for unsliced skirt steak). Add meat to board and let rest about 5 minutes.
Meanwhile, warm a few tortillas at a time over cooler side of grill, turning once, 30 to 60 seconds total; as warmed, stack and wrap in a towel.
Thinly slice meat across the grain into strips, saving any juices to pour on top. Cut cactus into strips. Set out meat and vegetables with tortillas and salsas, for making tacos.
*Find thinly sliced skirt steak or flap meat at a Latino meat market or supermarket (it may be called arrachera or ranchera). Find whole cactus pads and spring onions (cebollitas, which have large bulbs) at Latino markets and well-stocked grocery stores. To remove spines and fuzzy dots from cactus, wear gloves and scrape a knife down flat sides; also pare rims. Don't rinse, or cactus will get sticky.
Make ahead: Through step 2, up to 1 day, chilled.