I’m running Arch on the current latest rolling kernel (6.8 something I believe) and also use a Razer Core X with my Framework 16. I do not have the issues you’re reporting.
You don’t need to switch distros to use a different kernel, though I’m not sure what the user experience is like on a debian based distro like Mint for using more up to date kernels.
If it doesn’t boot with the eGPU disconnected, your issue may be something else.
eGPU related thread of mine here:
Are you connecting display output via the eGPU? I had issues getting that to work reliably and found it easier to simply connect through the FW itself and take the performance hit (which seems to be negligible).