The enzymes in kiwifruit and pawpaw are very strong and will destroy texture if left for too long. Good lamb should not require this treatment. I have a Greek friend whose father (now in his 90's) cooked the best lamb skewers you can imagine. A very simple marinade of olive oil, lemon juice, white wine, garlic, salt and oregano. Don't cut the meat too small. 1 1/2" cubes is ideal. Lamb cubes will shrink a lot if overdone and people do have the tendency to overcook. It will take on a beautiful flavour if cooked over hot coals like heat beads or lump charcoal. It needs to be close enough to the heat which scorches the outside leaving the middle pink. Fat dripping from the lamb onto the coals will cause flame ups and this is great for flavour. The bigger your cuts the further from the heat and slower cooking. If your lamb is too lean you may need to add some fat between the cubes.
Onion juice, tomato puree and lemon juice will all tenderise meat and go great with lamb. Salt, pepper, red peppers, chillis and of course garlic all go great with lamb. I would avoid yogurt, but I have read that milk is sometimes used in Turkish marinades.
12 years ago, for my eldest daughter's 18th birthday, I made a souvla grill / charcoal barbecue just for lamb skewers and cut, marinated and cooked 5 legs, but the best lamb skewers I've ever eaten were cooked on a $20 hardware store Hibachi BBQ using heat beads and served with a squeeze of lemon juice and pinch of salt.