However, it doesn't seem like Legacy likes AMD that much, I've read somewhere that it has driver issues which makes the game unplayable. I've encountered it before—the load times can get very long and sometimes it gave me an Orange Sidewinder.
I'm not aware of any serious issues with Legacy and AMD drivers. There was an issue with terrain generation crashing with Orange Sidewinder errors in Odyssey, but that was resolved years ago.
If the issue is really on the PSU, then my system would shut down no matter the version, right?
Not necessarily. There are significant differences between the 3.8.x and 4.0.x renderers.
Eliminating driver (I suggest the Adrenalin 24.5.1 drivers, for now) and other configuration issues would be wise.
You can run something like
OCCT's power stress test to see if you can induce any issues.
Odyssey rendering is, compared to legacy, power hungry - there's a lot more happening on your GPU.
On most of my cards, I can actually get Legacy to pull slightly more power. Watts per frame is significantly lower, but frame rate is also significantly higher.
However, I have no realistic way of accurately measuring peak transient power draw. Software averages from on-board monitoring ICs don't have particularly fine temporal granularity and I don't have a quality occiliscope.
My suspicion is that your PSU can't provide the required 150 W over the GPU power cable (seeing that both Gigabyte RX 7600 models have one 8-pin connector), but maybe only 75 W (the older standard). This means that the ~160 W power consumption of RX 7600 is right on the edge of your system's capability to power it (75 W from PSU, another 75 W from the PCIe slot) and the PSU's overcurrent protection triggers when your GPU tries to pull the maximum power it needs.
I'm highly doubtful that the issue is at the cables or connectors, unless something isn't seated correctly (something to double check) or is seriously damaged. Even a single 8-pin PCI-E connector with three 18 AWG +12v wires can safely deliver way more than the 150w spec, sustained. There is a reason why 2x8-pin to 12VHPWR adapters/cables that fail usually fail at the 12VHPWR end...2x8-pin connectors rated for a total of 300w can typically carry more power than a 12VHPWR connector rated for 600w, because the former typically has ~150% margins, while the latter has ~20% margins.
The issue is very probably with transient spikes, likely at the PSU, but possibly at the card itself. Could also be software, but with almost any remotely recent driver, none of the 7000 series cards should be having any unusual issues with terrain generation and even if they did, it wouldn't be causing the system to power off.