OMG. Would you believe it.
So, I splurged a bit and got myself a new 4K monitor worthy of my new system, which actually has a DP port, unlike my old screen. I also bought a DP/DP cabel for it - I had a tiny suspicion that the passive DP/HDMI combined cable might cause some issues (I mentioned it in passiing in a previous post), so I just plugged the live Solus USB stick in and booted.. lo and behold, the system came online without any further action on my part whatsoever!
I am now an owner of a system running Linux Solus {Budgie) - so proud!
Thanks everyone for both the literal and moral support and - as far as I can tell, the cable/ monitor was the issue. Certainly not the first thing one thinks to tamper with when dealing with OS boot issues, but there we go 🙂