On the other hand, the human mind is Turing complete.
This is a common confusion about what "Turing completeness" entails. The human brain is not Turing complete, as it is not infinite and doesn't operate on well defined rules (that we have totally cataloged). It might be equal to a linear bounded automaton. Turing completeness applies to rule sets, not the machines that run them. It's commonly applied to languages. Less technically, some people will say "Turing complete" to mean that it can do arbitrary or human-like computation, but this is not the true definition link.
To your question:
Why would we expect any of these AI paradigms to give us human level intelligence (AGI) if the former is not Turing complete while the latter is?
Assuming you're using the lay definition of Turing completeness, Current AI algorithms are nowhere near close to what the human brain can do, and everyone in AI that I know understands that radical, as of yet unknown, advances in order to truly approximate the human mind.