I'd like to point out the "breadboard psu" link in the first post is a really bad one.
It only has one (probably low capacitance and very cheap) capacitor in the circuit design.
This is not good for AMS1117 (or friends) linear regulator.
That regulator needs both input and output capacitor in a normal design.
To use it for an Omega 2, these capcitors have to have high enough values (I suggest 220 uF.)
For the bursty-power-demand device such as Omega 2,
you really cannot cut corner in this front in power supply department.
All that can be said is that your current measurement does not by itself indicate much.
If you could log instantaneous current over time and compare to the startup of a functioning board it might be possible to learn something. And that other person who measured more than an amp had in their measurement a clear indication of a problem, but your measurement is not outside the expected range, at least until its trend over time is considered in detail.
Note that it is not out of the question that connection of an unspecified ammeter inline could prevent the system from properly functioning.
Relatively unlikely to make a difference, by now that you've done the upgrade you might try booting it without the USB stick plugged in - it represents some additional load on the power circuitry, and adds some complexity of software possibilities.
Generally speaking there are three categories of things that could be wrong
You could have installed a bad image, or at least one that's wrong for the board. Always failing in the same place in the boot sequence would be a hint of this, but it would seemingly not occur with a different image especially one that previously worked.
You could have a power issue that's causing memory corruption rather than an immediate reboot. Always failing in a part o the boot process that involves wifi would be indicative of this
You could theoretically have a badly soldered board or one with bad components, not so bad that memory detection fails right away in U-Boot as has happened in a small number of cases, but more "flaky". However if a boad-level issue this would seem likely to cause more random failures than specific ones. An actual bad memory cell could cause a consistent failure but seems yet less likely.
@Chris-Stratton The problem is, that it does not reboot. It stucks with that message!
There's a chance the CPU did reset, but was unable to read the flash chip because on boot it would be sending 3-byte addresses while the flash chip may still be expecting 4-byte ones from the settings in use before the crash.