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It doesn't mean everything, but it's more reliable as an indicator of talent than any other single indicator.
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That is a huge overstatement. I know the AIME very well. I helped my daugther prepare for it. It has very little to do with true knowledge of mathematics: very basic algebra, some geometry, no calculus, no analysis, no proofs.. At least the USAMO and IMO involve proofs and some combinatorics and some very challenging problems. A USAMO or IMO MEDALIST is clearly showing talent in mathematics. I studied with many very strong mathematicians and several went to MIT for PhDs in math where they excelled. None ever participated in the USAMO or IMO but easily outperformed many others who did. While I was good in math in college, I was nowhere near their level. Still, 30 years later, I still get an 8 to 10 on the AIME under timed conditions. Is it an indicator that I have the talent to become a good mathematician? Hardly! It just shows that I am just very good at logical puzzles. It is also a highly trainable skill. I got my daughter from a 2 to a 5 in one month of practice after she qualified for the AIME. Is it harder than speed Sudoku, cubing or solving the Sunday Times crossword puzzle? I don't think so. It does provide some measure of logical skill ability, beyond what the SAT can measure. But it is NOT a measure of true mathematical talent and certainly not a measure of who will be among the top achievers at MIT. If we are looking at some indicator of future performance at MIT I would venture to claim the top finalists on the science fairs such as the Intel or Siemens competitions do much better than most USAMO finalists. Same thing for those who get a research paper published. At least they have to create something from scratch even if they often get some help. These are long term projects that truly require a passion for science.