The biggest challenge here is getting the crushed ice; if you have a really good ice maker in your fridge (and not many people do), use that. I tried blitzing my ice in my high speed blender and it nearly pulverized the cubes back into water. My suggestion? Take our your frustrations in a positive way double bagging few cups of ice in a ziplock, getting your nearest meat tenderizer, rolling pin, or hammer and letting ‘er rip. You want crushed ice, so once it gets to the size of pellets, you can stop. Unless you’re really pent up, then keep going, but get new ice for the drink.
Alternatively, if you don’t want to mess around with the ice situation, another method that can work well is the “granita" method. Blend all the ingredients together in a clean loaf pan and place in the freezer. After about 90 minutes - 2 hours, take out and scrape the slightly frozen mass with a fork, scraping it into a snowcone-like slushy consistency. Place back in the freezer to freeze completely, then spoon into a glass when ready.