So after much fussing around and hair pulling, I must admit I had accidentally stumbled upon the solution.
The issue I was having was not the odd hardware or some technical failure (RAM sticks are healthy and removing them or disabling LAN did nothing to help) - it was the fact that I was trying to install ESXi host without and SATA disks attached at all. As soon as I attached the SATA drive I just happened to have handy, everything worked flawlessly.
This was a completely accidental discovery - in a particular moment of desperation, I tried to run CentOS 7 install ISO in an effort to install and boot it from USB flash drive (hopefully to go even deeper and do additional diagnostics). It failed with exactly the same symptoms as the ESXi except - it threw a message akin to "no controller attached". That immediately got my attention as this is the latest stable and server-grade free distribution that isn't supposed have to have any issues. Therefore it must be the kernel itself (which is probably the only thing ESXi distro and CentOS7 have in common) is behaving erratically when no SATA drives are connected. Helped by the message (which was lacking from ESXi efforts), I connected the HDD and - lo and behold - everything ran without a hitch.
My conclusion is this: for the reasons that escape me, it seems (stock?) Linux kernel is having major issues being installed on a system without SATA disk attached. Is it just this is manifesting in this fashion with only this particular motherboard or some other hardware component? I have no idea and did not want to test it any further - I am just glad it works right now.
On a personal note, I've been working with computers for the past 25 years and this has to be one of the top 5 bizzare/wtf moments I have ever had.
HTH someone.