No system is "perfect", they all have their limitations and weaknesses. As far as how good AEGIS or any other system is, the people who really know aren't allowed to say, so the only thing the rest of us can do is guess. My personal guess is that AEGIS is likely to perform better than Patriot because AEGIS has been around longer, is probably better developed and is the main radar and defence system in use by US navy ships. There is no backup if AEGIS doesn't work, so a lot of time and effort has gone into making sure it does work. What I can't guess it is how much better it's likely to be compred to Patriot.
If you read the link in my first post, the author of that article said that at the time when he was in the US navy it was unclear who would be hurt the worst in a massed Backfire raid against a US carrier, the carrier might be sunk, it might not, it would depend on the exact circumstances of the attack. The carrier group would lose some ships, that was for certain, but exactly how many wasn't certain. He was very confident that so many Backfires would be shot down that the same naval aviation regiment would not have enough Backfires left for a second attack (that wouldn't stop a different Backfire regiment from making a second attack though).
If a major war occoured between the US and China, there would not be just one US carrier involved, there would be at least two of them (my personal guess is that there will more likely be three carriers).