universality of quantum gates
Yes, approximate-any-quantum-gate implies can-solve-BQP-efficiently.

Even if the cost of approximating gates is very high, you only need to approximate Hadamard, CNOT, and $Z^{1/4}$ operations to some constant fidelity before error correction starts to work. So although you may pay a large constant overhead to get "good enough", once you get there the extra overhead to achieve better and better precision becomes reasonable.