I even went back to a previous safe, set him to "just be" on the terminal and left. On coming back, he was nowhere to be found. Seems like he's not supposed to stay around. Or it's just an ordinary bug.

See my post above yours. My understanding is it is a confirmed bug, he is there, he has just "despawned" assuming you have not sent him anywhere and there is no evidence of his demise.

My Dr. Feelgood die after being killed by a Deathclaw and then disappear.

Mine just disappeared. I'm going to try and un-disable him with the console when I get back. The terminal is still there - if it wasn't for the dead ghouls I'll probably be happier using that place. Right now feral ghouls, dead or otherwise, give me the creeps.

I think it's weird that I can drag that dead raider chick (poor girl spawned dead after some random encounter, and for some reason never despawned, and I felt bad that she was always lying there when all those other encounters spawned there almost everytime I returned) from under the bridge at Starlight Drive-In and place her in a nice secluded corner grave, and she stays there, while with the feral ghouls if I dragged them out of the co-op, they move back to where they were after some time passes. Maybe because the raider chick was a random encounter spawn. I shall try to drag Sal and Carl to Jamaica Plain settlement and see if they stay there. But yeah, those dead ferals ensure I never use the co-op even though I really like the layout. It's like a mini-Sanctuary where nobody gets lost.

The robot console is rather irrelevant until such time as Doctor Goodfeels can be stabilized so he does not just disappear or wander off.

I just experimented a bit with this robot:

I was one time at the Sunshine Tidings, and saw it. But at this time I decided I did not want another settlement, so I went away. In this case the robot disappears.

When I just returned the robot was not here, despite the fact, that I did nothing the last time I was there. But when I hacked the terminal and set the robot to 'Guard' he suddenly was there! He followed us (me and Curie) through the settlement and helped kill the ghouls. I think the devs wanted it just for lower level characters to help with the ghouls.

Afterward I set it to maintenance and it immediately set off. I ran after it - it has a very long way, down to DC, the along the railroad to Big Joes Garbage (?) and on towards Boston again. But at Big Joes a mutant hound killed it. I let it be that way - but I think at the shop (listed in the terminal) where the robot was stolen you could find it again, if you either protect it on its way or let it be so that it floats through unloaded cells.

I don't know which platform you're on. But on PC there's a mod that lets you scrap corpses.