The problem is that the two-diode model does not explain what you want it to explain - i.e. a transistor in saturation.
What the two diode transistor model does explain is the internal structure of a BJT in a very simplified fashion.
It explains that there are two PN junctions than can be viewed as two diodes, and if you individually try to use each PN junctions, it will look like two individual diodes.
In reality the model is flawed because you already seem to know that two diodes bolted together do not make a transistor. The two PN junctions are just arranged so closely physically that pushing current through BE junction makes such conditions (electric fields, moving current carriers such as electrons and holes) that allow current to pass through from collector to emitter too, based on base current.
So the model does not even try to explain how a transistor works in saturation, or in active mode for that matter.
Which basically the model means that if you have a multimeter that has a "diode test" mode, you can quickly check the BE and BC junctions separately to see if the transistor should still be OK or damaged. It can definitely be used to check if an unkown BJT is of NPN or PNP type, depending on which way the internal "diodes" are. To a lesser extent, it should also be possible to determine the pinout, as base will have the common terminal, but the BE and BC junctions may have so small differences that it may or may not be possible to determine which way C and E pins are.