To be turing-complete, it must be able to simulate the function of every possible turing computation to infinite scale, in principle.
And other than a person getting bored, or forgetting something, or losing track of where they are, or running out of space or life, they can. Again, this is all that's required.
Sorry, but this is a math problem that you're thinking of like an engineering problem. It's just. Not. The point.
I've heard other people claim that if you can simulate any turing machine, you're a turing machine.
Yup, and they'd be wrong. It doesn't prove that. All we can say in that case is that a human being is at least as powerful as a Turing Machine, but can be more so as well. It's sort of like how you prove that Turing Machines and Lambda Calculus both solve the exact same set of problems. By simulating a Universal Turing Machine with Lambda Calculus you prove that it is at least as powerful as TMs. By implementing a Lambda Calculus evaluator with a TM you prove that TMs are at least as powerful LC. And together you prove they have the same computational power. The people you're mentioning with their claim are missing part 2.