Please don't mix up things.
At the moment I only see one issue with FlashMagic for the LPC54628: reading the first 4 bytes from internal flash sector 0.
This is due to a restriction in the ISP ROM code in the LPC546xx devices. Below is the place in the UM where this is mentioned:

The difference between reading with Flash Magic and a debugger is described below:
- I use for testing a correctly running LPC54628 board
- I don't change the memory content, I don't erase or program anything on this board
- I read out flash sector 0 with Flash Magic
- I read out flash sector 0 with a JTAG debugger (SWD), no matter what sort of debugger

Jonnie, if you can run your boards when starting them from Flash Magic but not without, then you may have forgotten to apply the vector checksum value at address 0x1C to the program binary image (check information to this in the User's Manual, chapter 3.3.2.7).
So please don't start changing the MCUs on your board before you didn't clarify this. I'm rather sure that the MCUs are ok.
Flash Magic will write a binary correctly to flash memory, from what I see it is just the read-out function on this LPC54000 MCU architecture which does not allow for reading the first word and therefore FlashMagic reports this word as 0x0000 0000.
Regards,
Bernhard.