It won't be a cheap backup if you calculate how much of a heatsink you'll need for this. You'll be dissipating 21W in those regulators. You'll probably want a 2C/W heatsink at most. That's an easy $25 right there + machining to get the threaded holes to mount the regulators. Are you telling us that you can't get a second 12V->5V 3A converter in parallel as a backup for less than that? I don't believe it for a second.
One idea I've come up with is to have a current-equalizing "current mirror". It auto-selects the highest of a variety of currents to be the reference current. If the outputs of the mirror all drive the same current sink, then the currents will equally distribute among the branches. It's not a high-precision circuit, but it doesn't need to be: there should be plenty of derating applied to any high-current regulator circuit.
This works with any number of regulators, and it also splits the heat dissipation between the regulators and the pass transistors. It has the drawback of requiring about 0.5V-0.7V of additional dropout voltage.
The partitioning of dissipation of each regulator channel between the pass transistor and the regulator is uneven. For the regulator with lowest output voltage, the regulator has most of the dissipation vs. the pass transistor. For other channels, the pass transistors take most of the dissipation.
This circuit is fail-safe: if any of the regulators fails open, the output will largely shut-down.
I expect the transient performance on increasing load current to be worse than when the load current drops off, since all but one regulator are working at the drop-out threshold, and their regulation in that operating regime is not as great for rising loads.
simulate this circuit – Schematic created using CircuitLab
The Schottky diodes select the regulator with the lowest output voltage to be the current reference. R1's value is chosen to provide enough base drive for the lowest input voltage and highest output current. C1 provides a soft-start. The pass element transistor type can be chosen for low Vce at the desired operating current.
This circuit also largely doesn't require the voltage regulators. The regulators, if present, act as per-channel protection devices, with an inherent "wire-or" shutdown for the whole supply if any regulator's protection trips.
If the per-channel protection afforded by the regulators is deemed unnecessary, the regulators can be removed and replaced with a piece of wire from IN to OUT. Only one of them has to be actually a voltage regulator. The voltage regulator, if mounted among the pass transistors, will act as a temperature limiter for all of them, and due to the current mirror action, will also current-limit all the channels in parallel.
Since this then becomes a single-input mirror, the Schottky diodes and R1 can be removed. The reference channel - with the regulator - needs to have the usual collector-to-base connection, perhaps with a base current compensation resistor thrown in.