I fixed the boot !
root@solus / # efibootmgr -c -d /dev/sda -p 1 -l "\EFI\systemd\systemd-bootx64.efi" -L "Linux Boot Manager"
efibootmgr: ** Warning ** : Boot0000 has same label Linux Boot Manager
BootCurrent: 0004
Timeout: 1 seconds
BootOrder: 0006,0005,0003,0004,0001,0002,0000
Boot0000 Linux Boot Manager HD(1,GPT,76eadfc2-43c4-4317-b7db-4d28d1188964,0x40,0xf4240)/\EFI\systemd\systemd-bootx64.efi
Boot0001 Hard Drive BBS(HD,,0x0)
Boot0002 CD/DVD Drive BBS(CDROM,,0x0)50333a20484c2d44542d53542044564452414d20474832324e533730202000
Boot0003* Solus Linux Bootloader HD(1,GPT,76eadfc2-43c4-4317-b7db-4d28d1188964,0x40,0xf4240)/\EFI\com.solus-project\bootloaderx64.efi
Boot0004* UEFI: Corsair Voyager GT 3.0 000B PciRoot(0x0)/Pci(0x1c,0x4)/Pci(0x0,0x0)/USB(2,0)/HD(1,MBR,0x5bedc865,0x88,0x64000)
Boot0005* Solus Linux Bootloader HD(1,GPT,76eadfc2-43c4-4317-b7db-4d28d1188964,0x40,0x1f4fc0)/\EFI\com.solus-project\bootloaderx64.efi
Boot0006* Linux Boot Manager HD(1,GPT,76eadfc2-43c4-4317-b7db-4d28d1188964,0x40,0x1f4fc0)/\EFI\systemd\systemd-bootx64.efi
root@solus / # clr-boot-manager update
[ERROR] cbm (../src/bootman/update.c:L189): Cannot determine the currently running kernel
The issue was indeed due a broken UEFI partition I was not able to fix while booting a non UEFI live session.
now I still have issues with my GPU but I can SSH in, so that will be for another topic, if any.
Thank you for your feedbacks and support, and for the fixing boot guide to whoever edits it, it has been a great help !