I don't think I know a human who would have a zero error rate doing 1,000,000,000 Turing operations
Nor do I. However, this is irrelevant. In determining whether a system is Turing complete, physical limitations are usually ignored. From Wikipedia:
To show that something is Turing complete, it is enough to show that it can be used to simulate some Turing complete system. For example, an imperative language is Turing complete if it has conditional branching (e.g., "if" and "goto" statements, or a "branch if zero" instruction. See OISC) and the ability to change an arbitrary amount of memory locations (e.g., the ability to maintain an arbitrary number of variables). Since this is almost always the case, most (if not all) imperative languages are Turing complete if the limitations of finite memory are ignored.
If we did not ignore physical limitations, no actual computing system would be Turing complete.
Making errors means not behaving as a Turing machine. It's separate from limitations of memory.
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