Sheemie is a dude
Sheemie was controlling the machine with his powers. You see it right in the scan that he's still psychically controlling it by the fact that it's glowing with his psychic energy, and if you really want to go into the details, it still would have been incapable of moving on its own if Sheemie had done nothing but jumpstart its circuits again. This is all but stated to be the same robotic horse that got its head sliced off by a light-stick in Wolves of the Calla.
Moreover, I haven't even gone into how Dinky Earnshaw (another psychic weaker than Flagg) mentioned how easy it was for psychics like him to actively burn out the Old Ones tech that the Warriors of the Scarlet Eye were using. And as for how Earnshaw ranks on the scale of his verse, the guy is on the same level as Ted Brautigan, who considers Sheemie's powers to be outright remarkable.
Flagg has a history of simply using his hax against people, arrogance be damned. He turned a man into a dog just for annoying him, turned three gunslinger officers into dogs when they attempted to arrest him, threatened to do the same to a few subordinates just because they were slacking off, regularly uses command-attacks against people who can't hope to defeat him anyway just because it's a fast way of taking them out, blasted Alain Johns out of a lucid dream just for attempting to rescue Roland from inside of it, and drove a man insane because he messed up one job. One of the people he mindhaxed was literally mentally handicapped, two others were children, and a third and fourth were Jake's completely unremarkable parents. Arrogance means nothing when the man's history is full of examples of him just not caring in the end. There are reasons that Roland & co. are still alive after meeting him, and in this case it's not because of his own hubris. And even then, SBA = in-character, but willing to kill, so this shouldn't have been brought up to begin with.
He's also not stupid. In fact, he's probably one of the most intelligent non-cosmics in Stephen King's mythos, and as shown quite a few times, he certainly knows the limits of his own powers. Soon as he's faced with a literal robot he has to fight, he's not going to bother with telling it to "stop breathing" or hitting it with a flash of light or making some other rookie mistake. He's going to do the natural thing and either take control of it or transmute it. Likely the former.
Flagg needing to affect Hatchworth's core in order to transmute him has plenty of logical holes, but the biggest one that I can see is...why would he need to transmute the core when transmuting the exterior would be enough to incapacitate? Even if the core stays the same, the body is unusable.
Same with "overpowering" Hatch's core in order to shut him down. On general principle, you don't need to "overpower" a machine's batteries in order to shut it off, and unless you can provide proof that he's a special case in that regard, said principle applies here as well. Even then, thanks to lack of resistance, Flagg taking control of Hatch means he can't do anything, and if he really needs to put the robot down, he can blow out his individual components just by speaking.
You still haven't provided any proof that Hatchworth's alternate selves will somehow know out of instinct (or whatever) that this particular version of him is in need of any assistance. All the song's lyrics show is that he has an alternate dimension in his core and that he can time travel, not that he can use those abilities to bring new versions of himself into his time period while he's not even functional in order to do so.
In the end, it boils down to Flagg's psionics letting him manipulate machinery as he pleases, weaker psychics easily doing the same, Hatchworth having zero resistance to such, Flagg having the mindset to go right for the good options, and Hatchworth losing in a quickdraw. That's my vote, and I'm sticking with it.