Heisenberg's approach was intuitive first. He thought about, what's really observable concerning an atom (particularly the hydrogen atom which was the only one which seemed to work with the plain Bohr-Sommerfeld quantization), and he came to the conclusion that it's the spectral lines, i.e., the transition rates between energy levels ("orbits").
This brought him to develop a formalism for the the harmonic oscillator first, leading him to a quantum mechanical "reinterpretation of classical observables". He also developed an algebra for his scheme, and Born immediately realized that this (non-commutative) algebra is just the algebra matrices obey, but that one needs an infinite-dimensional matrix.
It's of course natural that this version of quantum mechanics, developed by Born, Jordan, and Heisenberg quickly after Heisenberg's heuristic breakthrough, was formulated in what we nowadays call the Heisenberg picture of time evolution, i.e., with the full time dependence on the operators (matrices in their matrix-mechanics formulation).
The hydrogen problem was solved by Pauli within matrix mechanics.
Of course, everything became a lot more easy to handle with Schrödinger's wave mechanics, and after a big fight between him and Heisenberg, which theory might be the correct one, Schrödinger proved the complete mathematical equivalence. The most elegant formulate, of course, was Dirac's general "representation free" formalism, showing that everything can be formulated on an abstract Hilbert space and operators acting on it.