But what I don't grasp is why sharing them causes a problem.
Because it pushes more current through the resistors and so the resistors drop more voltage giving less voltage to go across the LEDs.
Does R3 also have a greater voltage drop?
Yes but you only get this once. The middle two get it twice.
The voltage at the hot end of R5 is going to be higher than the voltage at the hot end of R1, so this backs off the voltage across L2.
Anyway I thing you have answers the original question, will this work, it's a no.