Given my extreme laziness what's the benefit of 31m/s in vertical velocity? I guess one would wag the nominal energy required to get to orbit and then 1/2mv2 a 0th order fractional energy offset from the huck-o-gizmo?

The rocket equation is cruel, particularly when combined with gravity losses. Saturn V used up 8% of its fuel just clearing the tower. So a little boost at launch can go a long way. It still may not be worth the added complexity, but I bet they've done the math. If the arms are reliable enough to catch the booster, they're probably reliable enough to throw it (or rather, to give it some assist on launch; even a 10% effective weight reduction is huge during the first few seconds).

My go to for the rocket equation:

Rocket Equation Calculator
If the full stack is 5,000 mT and the isp is 330, then the first 31m/s would require burning off 48mT of fuel. 38m/s takes 53mT.

However, that is the in-orbit calculation (ideal). Since the rocket is fighting gravity, the acceleration and fuel burn depends instead on total engine thrust and burn rate.

Number of engines: 28

Full stack mass of around 5,000 mT (metric tons)=5,000,000 kg = 5Mkg

Total thrust: Raptor is 2,200 kN * 28 engines = 61.6 MN. (Might be 65MN)

Force due to Gravity is 5Mkg * 9.8 = 49 MN.

Net force 12.6 MN.

Acceleration = 12.6 MN / 5Mkg = 2.52 m/s or about a quarter of a G.

Assuming mass stays relatively constant for ease of calculation:

Fuel consumption: 565kg/s

Note that using a straight 1:1 counterweight approach would result in only 9.8 m/s acceleration until the rocket hit a thrust to weight ratio of >1. Anything higher would out accelerate and unload the counterweight.

Edit: need mechanical advantage in addition to the mass:

~~Using a 1.25x mass counterweight would provide the acceleration calculated here.~~
With a 1G addition: 9.8m/s + 2.52m/s = 12.32m/s

50 m travel: t=sqrt(50*2/12.32) = 2.85s

Speed at end of boost: 2.85 * 12.32 = 35.1m/s

Time to speed without boost: 35.1/2.52 = 13.93s

Time saved: 13.93 - 2.85 = 11.08s

Fuel saved: 11.08 * 565 * 28 = 175Mkg or 175mT

This does not correspond directly to payload, and they would not rely on the boost to get to orbit, but it would increase the amount of fuel Starship or Tanker have once they get there which is the critical factor for refueling trips needed.