Hi there, I just thought I have the same problem. I have installed solus on an external USB SSD. I boot it on a notebook with an Intel GPU as well as on a PC with an AMD GPU. On both devices I had no GUI.
Then looked into lsmod and saw there's no amdgpu loaded. After that I saw it bootet an old kernel 4.9.210 although I don't have linux-lts installed. I tried clr-boot-manager update but It still booted kernel 4.9. I had to manually select 5.6 on the boot menu. Now I removed the old kernel from /boot/EFI/com.solus-project. Now it's working fine.
I don't know why there was still this old kernel in the boot folder. Just in case someone has the same problem.