I have a Kitchenaid stick blender and I've made smoothies with it most mornings for the last 10 years. It does totally fine with frozen bananas, berries (including strawberries), and cherries. It struggles a bit with big chunks of frozen peaches and pineapple and mango because they're harder, but it still gets the job done. The only thing I had to pull out the big blender for was pomegranate seeds in one of the frozen fruit blends I bought. The stick blender did NOT like those, lol. I've never tried it with ice cubes, though, so you'd have to see.
One tip is to put the hardest things in the middle of the cup as you're layering everything in. I do yogurt, then spinach, then frozen fruit (hardest), then fresh banana, then flax/chia/protein powder. Then top it off with almond milk and blend. It's also important to work relatively quickly. If you get distracted for more than a minute or two between pouring in the almond milk and actually blending, it can kind of freeze with the berries and make a big clump that's harder to blend.