Unsolvable Incompatible Equations of Modern Physics = Complete Success!

All books on modern physics start out with a praise to Schrödinger's equation of quantum mechanics and Einstein's equation of general relativity as the highest achievement of science. Here is what Stephen Hawking says in The Grand Design (while signaling that something is weird):

• The quantum model of nature encompasses principles that contradict not only our everyday experience but our intuitive concept of reality. Those who find those principles weird or difficult to believe are in good company, the company of great physicists such as Einstein and even Feynman, whose description of quantum theory we will soon present. In fact, Feynman once wrote, “I think I can safely say that nobody understands quantum mechanics.” But quantum physics agrees with observation. It has never failed a test, and it has been tested more than any other theory in science.

Then comes a little caveat saying that unfortunately the equations are incompatible, and so one of them must be wrong, but in any case both equations are certainly valid/true and since both are written in stone none of them can be wrong, after all. Here is what Wikipedia says about the situation and the  prospects:

• Through years of research, physicists have experimentally confirmed with tremendous accuracy virtually every prediction made by these two theories when in their appropriate domains of applicability. In accordance with their findings, scientists also learned that GR and QFT, as they are currently formulated, are mutually incompatible – they cannot both be right.
• Some theoretical physicists currently believe that a quantum theory of general relativity may require frameworks other than field theory itself, such as string theory or loop quantum gravity. Some models in string theory that are promising by way of realizing our familiar standard model are the perturbative heterotic string models, 11-dimensional M-theory, Singular geometries (e.g. orbifold and orientifold), D-branes and other branes, flux compactification and warped geometry, and non-perturbative type IIB superstring solutions (F-theory).

So string theory is the only hope, but the string hype seems to be fading and so the prospects seem pretty dim. But there is another even more cumbersome problem with these equations: They cannot be solved!

Schrödinger's equation for an atom with $N$ electrons is formulated in terms of a wave function which depends on $3N$ space coordinates, which makes numerical solution impossible for N > 10
according to Nobel Laureate Walter Kuhn, and already the case N=2 of Helium is filled with difficulty not to speak about the case of Oxygen with N=8. And analytical solution is known only in the case of Hydrogen with N=1.

Einstein's equation is even more difficult to solve and only a few analytical solutions in extreme simplicity are known (e.g. vacuum solution of a spherically symmetric gravitational field for a static mass),  and numerical solution is not really an issue because data such as initial and boundary values and forcing are completely up in the air and the choice of coordinate system is unclear. We quote from Baumgarte Numerical Relativity: Solving Einstein's equations on the Computer:

• Chapters 12 and 13 focus on the inspiral and coalescence of binary black holes, one of the most important applications of numerical relativity and a promising source of detectable gravitational radiation. These chapters treat the two-body problem in classical general relativity theory, and its solution represents one of the major triumphs of numerical relativity. 

We understand that modern physics is based on two equations which cannot be solved, except in very simplistic cases. Yet the equations are claimed to be true in the sense that solutions of the equations always agree with observations, in all cases including complex cases.  But wait, how can you know that solutions of the equations always agree with observation if you cannot solve the equations and produce solutions to compare with observations?

Of course you cannot know that. But this troublesome fact for modern physicists, is twisted into: Since solutions cannot be computed, it is impossible to find any discrepancy between predictions according to the equations and observations! There are no predictions from solving the equations and thus there is no discrepancy!

More precisely it works this way: Suppose you have computed a what you view as an approximate solution of Schrödinger's equation for some atom with many electrons, by an ingenious choice of "atomic orbitals" combined with some optimisation to choose a best combination of orbitals, and that the predicted energy is in perfect agreement with observation. Then you congratulate yourself and say that you have produced yet another piece of evidence that solutions of Schrödinger's equation always agree exactly with observations. On the other hand, if your approximate solution does not agree exactly with observation, then you blame the approximation and take that as evidence that without approximation the agreement certainly would be complete, and then you try some other orbitals...until complete agreement...possibly by twisting the observation under the firm conviction that solutions to Schrödinger's equation give an exact description of atomic physics.

The net result is that the unsolvable equations of modern physics, which unfortunately are incompatible, anyway both must be valid to an unprecedented precision, since there are no examples of slightest discrepancy between prediction based on solving the equations and observation.

In other words the equations serve like oracles who know exact answers to important questions, but are not willing to reveal the full truth. Not very helpful.

Do you buy this? Or is there something fishy about solutions to unsolvable mathematical equations, which always give  results in perfect agreement with observations? Doesn't perfect agreement sound little bit too good?

Some quotes, among many similar:

• In fact, it is often stated that of all the theories proposed in this century, the silliest is quantum theory. Some say that the only thing that quantum theory has going for it, in fact, is that it is unquestionably correct. (Michio Kaku)
• When thinking about the new relativity and quantum theories I have felt a homesickness for the paths of physical science where there are ore or less discernible handrails to keep us from the worst morasses of foolishness. (Sir Arthur Stanley Eddington)
• Einstein, my upset stomach hates your theory [of General Relativity]—it almost hates you yourself! How am I to' provide for my students? What am I to answer to the philosophers?!!(Paul Ehrenfest)
• I count Maxwell and Einstein, Eddington and Dirac, among “real” mathematicians. The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are at present at any rate, almost as “useless” as the theory of numbers. (G. H. Hardy)
• Quantum field theory, which was born just fifty years ago from the marriage of quantum mechanics with relativity, is a beautiful but not very robust child. (Steven Weinberg)
• Niels Bohr brainwashed a whole generation of theorists into thinking that the job of interpreting quantum theory was done 50 years ago. (1969 Nobel Laureate Murray Gell-Mann) 
• Let me say at the outset, that in this discourse, I am opposing not a few special statements of quantum physics held today (1950s), I am opposing as it were the whole of it, I am opposing its basic views that have been shaped 25 years ago, when Max Born put forward his probability interpretation, which was accepted by almost everybody. I don’t like it, and I’m sorry I ever had anything to do with it. (Erwin Schrodinger talking about Quantum Physics) 
• One might very well be left with the impression that the theory (of general relativity) itself is rather hollow.: What are the postulates of the theory? What are the demonstrations that else follows from these postulates? Where is the theory proven? On what grounds, if any, should one believe the theory? ....One’s mental picture of the theory is this nebolous mass taken as a whole.....One makes no attempt to derive the rest of the theory from the postulates. (What, indeed, could it mean to “derive” somtheing about the physical world?). One makes no attempt to “prove” the theory, or any part of it. (Robert Geroch in General Relativity from A to B)