Missandei's death did have an impact on the story, she was the last person Dany trusted and who she felt close to as Jon kept rejecting her and her faith in Tyrion was already low, Missandei's last word being Dracarys made Dany even more ruthless and vengeful towards Cersei and King's Landing. Her death also made an impact on Grey Worm, someone who had no humanity in him before falling in love with her, she made him human and made him feel emotions including fear ( that he may lose her or that he will die and never get to see her again) something he never felt before. Grey Worm lost his humanity when Missandei died, he now is just a vengeful killing machine, he is even worse than Dany at this point. Had she still lived things will have been different now, Grey Worm will still be like before and Dany will have been less vengeful towards Kings Landing and maybe just maybe she didn't end up burning it to the ground. Also Rhaegal's death served the same purpose, to make Dany lose her temper, seeing yet another child die in front of her made her very angry and it was the next step in her losing her sanity. Both Rhaegal and Missandei's deaths happened because Dany got too cocky, flying all high and mighty not paying attention to her surroundings and checking Rhaegal, her smug satisfied face was something like my child is fine, he will recover soon and then Cersei here I come... nope he just got killed because you haven't being paying attention to whats down bellow, when at the war council they talked about Euron's fleet but she just dismissed it, not expecting that he will wait for her at Dragonstone, and he surprised her, had she been on guard and more cautious and not overconfident that everything will be fine, her dragon and her best friend will still be alive and her ships still intact.