Sorry for the delay in replying.
Putting the lassi in the fridge is just to cool it. Some crushed ice would probably make it immediately ready to drink but would water it down. The mango pulp gives it more flavour. I've done a few experiments over the past few days and have actually arrived at a final recipe that produces a perfect mango lassi - at least, it's perfect in my eyes but may not be in yours. I played around with just adding more milk but that turned the taste into something just like a milkshake. Not that there's anything wrong with milkshakes but I don't like them too much and need to disguise/hide the milk taste.
The final recipe is as follows:
250g unsweetened yogurt
200g mango pulp
1 cup milk
1 cup water
Seeds from 3 green cardamom pods
Just put everything into a food processor and blend for 2 minutes. Then pour the lassi into 4 tall glasses and put in the fridge for at least 2 hours. You can stand a straw up in the lassi but still drink it through the straw.