The Kurzgesagt video is well produced but surprisingly misleading. The drawbacks it brings up are all based on launching all of our nuclear waste into space, using present-day rockets with today's reliability levels, dedicated entirely to this one purpose, and each carrying the largest possible payload of waste. Most puzzlingly, it dismisses hitting the sun as difficult - as if it's any harder than hitting the moon, Mars, an asteroid, or any other space object we've been hitting consistently for decades. "What goes around comes around" is literally a terrible oversimplification that ignores reality.

The impracticalities the video discusses simply aren't relevant to adding a modest amount of waste to rocket exhaust as I described. But yes I agree that the strawman concept of launching nuclear waste into space on the scale described in the video would be a terrible idea.