How to even start solving multi-diode problems?
Is there something I'm not paying attention to, that would greatly
simplify this problem?

Just start simply.
Ask yourself questions about (say) diode D2 which I've marked nodes on called A and B: -

The question you should ask is this: could voltage B be greater than voltage A. Think about it - is it possible that B is greater than A? If you conclude it can't be then you can remove diode D2 from the circuit.
Then ask questions about the likely voltages either side of D4 and simplify the circuit this way. What about D3 - could it be conducting current given its direction and position in the circuit?
Use your eyes and brain.