Is there strong supporting evidence of discrete electron shells or orbitals surrounding atomic nuclei? I realize the math works out and we have energy frequencies emitted, perhaps even atomic diameter measurements. But these still seem indirect and allow for other possible explanations. Is there direct experimental evidence for these? The orbitals concept is always shown almost as fact, not theory, so wondering what experiments must support such strong conclusions and don't mention any other alternatives?
Historically, the first strong evidence for the existence of discrete orbitals was the Franck-Hertz experiment. They threw energetic electrons to a gas tube (in their original experiment, they used mercury), and they found very sudden drops in the transmitted electron flux at concrete energies. Those events occurred when electrons inelastically collided with mercury atoms, and the mercury atoms only could absorb the energy in the collision at certain discrete energies. They also found that light with the right frequency was emitted after the collisions.
Since then, a really huge amount of experiments have been made that are even capable to manipulate atomic energy levels (for example, qubits based on the Jaynes-Cummings hamiltonian)
EDIT: To address the part of the discrete distances, I think that it's enough to invoke Coulomb's law. But I'll include a example of its consequences: the paramagnetism of rare-earth ions.
Paramagnetism is caused by the angular moment of unpaired electrons. Angular moment has two origins: rotation (i.e., orbital angular momentum) and intrinsic (i.e. spin). In many elements, like transition metals, unpaired electrons are located in the outer shells. In a crystal, these electrons are near the neighbor atoms, so they "feel" their electric field. As a consequence, their orbital angular momentum is affected, and can even be averaged to zero (it is said to be 'quenched'). Thus, for most metals, the magnetic moment comes only from spin. But in the lanthanides, the unpaired electrons are located in the 4f orbital, that it is very close to the nucleus, and the 5s and 5p orbitals are filled and they are farther (of course, I'm talking about the areas where the probability is larger, electrons are not deterministically located).
Those outer orbitals 'shield' the 4f electrons from the crystalline electric field, so they behave as if they were free electrons: their magnetic moments come from both orbital and spin angular momentum, and these elements have larger paramagnetic responses. That wouldn't happen if the 4f and 5s 5d electrons weren't located at different distances from the nucleus.