Debugging a nonsynchronous step-down power supply problem
If I understand correctly, you have two generations of PCB with the same parts but different layout. And the following result:


Gen 1 - no voltage drop
Gen 2 - voltage drops from 5 to 2.4 under load


If the components are the same, then the problem must be with connections. You may have an open or a short somewhere, or forgot to connect some component lead. You basically need to perform circuit analysis and determine what is different. Your Gen 1 proves that the circuit and components are capable of working, but then Gen 2 board has a flaw. My hunch is that the task of cutting traces and adding jumpers to correct a design flaw is incomplete: there must be (an) additional design flaw(s) you overlooked.

Debugging it needn't involve an oscilloscope unless you need to look at high speed signals in your circuit.


Power off the board and closely inspect all traces and connections. Does the Gen 2 board match exactly (electrically) the Gen 1 board?
Verify with a multimeter that traces are connected to places they ought to be, and that they aren't shorted to places they shouldn't. Ideally this would be done before populating the board with components, but you should still be able to confidently say that your ground plane isn't shorted to something else, and so forth.
Are you sure the components are the same? In populating the Gen 2 boards, is it possible you used the wrong value component somewhere which disagrees with the Gen 1 version? A transistor mounted incorrectly or a diode installed backwards can cause strange results.
With power on, there may be a component overheating because it is drawing excess current (and thus causing your supply voltage to drop). Leaving it powered on could cause further damage, but you might be able to quickly determine what component(s) are overheating (if any). Additionally, you might be able to make some quick voltage measurements with a multimeter to compare the Gen 1 and Gen 2 boards. Finding what nets are the same voltage in both generations could help you isolate a component or board section that is flawed.