So I did some college physics. Statics'n'dynamics. Engineer stuff. It ain't hard. Let's start with a model, shall we?
I'ma do that obnoxious engineering oversimplification thing where we model a horse as a sphere. I'm also going to start in Freedom Units and work backwards because ugh. So a lawn dart is about 12" long and weighs on the order of half of a pound. We're gonna simplify that into 30cm and one kilogram. We're also going to disregard the complexity of fins and non-uniform cross section and turn this thing into a highway flare, basically - let's model it as a cylinder 30cm long by 2cm in diameter.
LAWN DART = 1kg, 30cm x 2cm cylinder
We're going to do this because we're going to rig up a bola-thing out of jumprope and do some playground physics. Through sheer MacGyver awesomeness I'm going to craft a sling with a length of around 3 feet (1m) and I'm going to spin my lawn dart around my head at a rate of 3Hz. Once it's as fast as I can spin it (3Hz) I'm going to let go.
LAWN DART ANGULAR VELOCITY = 6pi radians/sec = 1080 degrees/sec
I'm then going to abuse the bejeesus out of online calculators because I'm lazy but will link them. In this case, I'm going to convert from angular velocity to linear velocity.
LAWN DART LINEAR VELOCITY = 18.85 m/s
Hey that's pretty good. I'ma ask my buddy Newton (Vf = Vi plus at) how long it's gonna be in the sky, assuming it goes straight up, because I'm lazy and because it's a lawn dart and that's what they're for, duh. No drag because again, lazy.
LAWN DART FLIGHT TIME = 1.92 s
Great now I can ask Isaac for its distance (d = v0t plus 1/2at^2).
LAWN DART DISTANCE = 18.1m
59 feet straight up. Bitchin'! Definitely more than people can throw. We got a soup goin' baby! But now we gotta get Isaac to run some of the harder numbers. Like, energy. KE=1/2mv^2.
LAWN DART KINETIC ENERGY (linear) = 177J
....shit. Why did the universe compel me to put a caveat on the KE? Musta been those two solid years of intro physics, followed by three solid years of engineering. Hmmm. KE = 1/2 Iomega^2. SHIT. Frickin' moments of inertia? I = L/omega. L = M x V x R. Uhhhm, M is mass is 1kg. V is angular velocity. R is radius around the centroid - hey we got all this shit!
LAWN DART ANGULAR MOMENTUM = 18.85 kg/m^2 s
I don't like those units. They're... foreboding. But now we can calculate the angular kinetic energy.
LAWN DART KINETIC ENERGY (rotational) = 3,349J
...that is more. more energy. Uhhhhhm somehow between spinning it around our arm and launching it straight up we bled off 95% of our energy. Which, my buddy Isaac reminds me, is not permitted by the universe.
So here's the basic problem: we were spinning that lawn dart around a fixed axis that happened to be "where I hold the jumprope." That lawn dart is not going to stop spinning just because we did. It's going to spin around its new system centroid. Good thing we already defined it as a road flare rather than a lawn dart 'cuz here's where stuff ceases to be fun (or starts, depending on one's perspective).
Now - I did this math? I did the shit out of this math 20-f'n years ago. So here's where I start fumbling. Because what we're worried about is conservation of angular momentum. ice skater pulls their arms in, they spin faster. In fact, if you look up "discus throwing momentum" you will get like long-ass papers that are mostly speculation, a little measurement and no real answers on the "conservation of angular momentum" problem. Because really, the momentum of the system is conserved, and we've presumed our rope (and me!) has no mass.
But let's fumble on and wait for Devac to poke holes in this 'cuz he's wicked smart.
Anyway: angular momentum be conserved, yo. We have our kinetic energy - it's 3349 - 177 = 3.2kJ. we can calculate our new moment of inertia to be 0.007556 kg-m^2. So through sheer dumb conservation of momentum, our lawn dart is now in a 146Hz flat spin.
Is it really, though? I have a degree in this shit and I'm pretty shaky on it. Turns out you physics nerds are better at this shit. Lo and behold you can't disregard half the system no matter how hard I, or Spinlaunch, wants to. But since we know even less about it than we do about the business end of the process, I'm not even going to try. I'm going to refer to that 146 Hz flat spin as the lawn dart's "suicidal urge" because while we can all agree that I'm not flippin' a lawn dart into a low thrum above my head, I think we can also agree that I'm not NOT putting a spin on it, and that when 95% of the energy is not pushing my lawn dart up, it's pushing something SOMEWHERE.
Note that "suicidal urge" isn't "jerk" but they're related. The suicidal urge is what changes the lawn dart spinning around my center of mass at 3Hz to trying to spin around its own center of mass at 150. Again - don't think it's 146, certain it ain't zero.
What we're looking at here, obviously, is KSP attempting to deal with corner-case physics it definitely wasn't designed for. I doubt it's got a 2-body problem algorithm in there. The launcher does not tear itself to shreds. It's got a vectoring motor capable of covering up nearly any crime. And even Kerbal is all "lol this thing is going through a full spin before we figure our shit out."
LAWN DART SUICIDAL URGE = 146Hz = 3.2kJ