How to install sane graphics drivers for Nvidia Optimus hardware on *Ubuntu 16.04?

I can see this becoming a recurring thing every few years.

Ubuntu 16.04 was released not too many months ago and I have bought a new hard disk and installed Kubuntu 16.04 on it. My laptop is something from a few years back with an intel something graphics driver on the CPU and an Nvidia GTX 670MX discrete card. (It's an optimus system.)

Now one used to get this working via some strange combination of bumblebee and nvidia drivers and editing text config files, etc, etc, you used to run primusrun/optirun executable_file and it (sort of, rather unconvincingly) worked. (Until a new update broke it again, or you wanted to use CUDA...)

See this question for additional context, and also see the questions linked there.

It is my understanding that the new release of 16.04 does away with all the primusrun/optirun stuff, and we now have a seemlessly switching graphics solution which works "much more better" than before. (Or is this just totally wrong information and I completely misunderstood what I read online?)

So, I attempted to do a sudo apt-get install nvidia-361-updates which brought along with it nvidia-prime... I rebooted back into Kubuntu and everything seemed to be working! My graphics card started to get warm and the fan started rotating, which doesn't usually happen unless the card is "on and doing something to make it heat up". I couldn't test it any further but I assume it was rendering stuff for the desktop environment.

But, somewhile later my system just suddenly turns off. I reboot and suddenly I can't login anymore... Either the system freezes, logs out or turns off.

So I attempted to follow information in this question.

Which just made things worse. Now I can't even get to a login screen. After boot my screen stays black/blank. However I can still get to a tty window.

There are other questions here and here which describe similar issues but did not resolve the problem. (I found these after searching for ubuntu 16.04 graphics driver blank screen or something similar to this.)

I've done the classic thing of trying too many things "which might fix it", but since I don't fully understand in detail the exact effects of all the commands I'm running I've got myself into a hole and it's probably time to start again with a fresh install and get it right the first time.

So my question is, how do I "get it right"? I have an optimus laptop with an Nvidia GTX 670MX along side an Intel integrated graphics processor, what combination of drivers, ppa's and software should I install on *Ubuntu 16.04, and in what order to make my optimus hardware functional?

In addition, am I correct or incorrect in my earlier statement when I suggest that something fundamentally big has been done to improve hardware support for optimus systems in 16.04?