I guess what I'm asking is this: Does the current monarch have control over who will succeed her?
Could Elizabeth decide to skip over Charles and William, and hand the crown over to Harry?
No she doesn’t. Those rules are laid down in the latest constitution of Great Britain. She certainly could abdicate in favor of the next in line, but that certainly would be a mistake. Charles could defer to his son, which he should do.
I’m of the opinion that the Queen should hang on till the end, which she’s indicated she’ll do. This blood line lives a long time on average, excluding her sister who smoked 4 packs a day and her father who did likewise. Edward the VII smoked twenty cigars a day along with many cigarettes during the last several years of his life and lived to about 69. Other than that they live longer than the average life span.