Assuming that the rocket has moment of inertia similar to a hollow cylinder with radius 1ft, conversion from rotation energy to linear is perfect, and final linear velocity is 5000 mph, the rocket would have to be spinning at
1/2 * I * w^2 = 1/2 * m * v^2
1/2 * (m * r^2) * w^2 = 1/2 * m * v^2
w = v / r = 7333 rad/s ~ 70,000 rpm
Maybe the payload of the rocket could be connected to the body of the rocket with magnetic bearings so that it isn't spinning, i.e. not experiencing the massive accelerations.
1) accelerate fast enough to penetrate the atmosphere at high speeds (and remember, the faster you travel, the air resistance is the square of your velocity).
2) withstand the crazy max-q as soon as you leave the spiral (which I assume is vacuumed).
3) make the centrifugal forces reasonable during the acceleration
The article mentions acceleration to 5000mph, which would be enough if there was no atmosphere, but I highly doubt it will work with.
Would it be possible and useful to do it with a 2 payloads system: one dumb piece of material first which can take lot of G followed closely by your "fragile" payload using its draft?
G-forces [1] are a consequence of acceleration. An accelerating object will experience g-forces in an atmosphere or in a vacuum. ("Drafting" [2] is an aerodynamic process by which a following object exploits a leading object's slipstream to reduce the former's drag.)
With a slingshot, the destructive force is the g-force inflicted by the acceleration. (With a rocket, the destructive forces are mostly vibrations and aerodynamic stress.)
Note that a slingshot doesn't require lots of Gs. One could use a super-long slingshot to achieve the necessary velocity. But super long is super expensive, so high Gs it is.
Very long cables are a lot cheaper than very large slingshots?
Anyway just thinking out loud.
Unless your cable were elastic, this would not materially change the payload's acceleration. That said, you might find skyhooks [1][2] interesting.
The problem is that now you have to transport everything near and up the mountain, which costs a ton in logistics (getting a road up a mountain that can handle a entire assembled rocket isn't cheap, nor easy). Most launches have delta-v to spare because satellites are usually nowhere near the rocket's maximum payload weight.
tl;dr it can save 6% in a ideal situation, but the extra cost and logistical problems aren't worth it.
At the proposed 5000mph, air resistance at sea level would be tremendous. Greater atmospheric losses would entail either bringing more fuel to complete the orbit, or larger launch loop / greater acceleration force.
Unless you launch by means of a mass driver/rail gun [1] built into the side of Mt Everest with a piggy backed elevator for tourists. Tourists would arrive to a pressurized viewing area at the summit and have the option of donning an oxygen mask, going outside and posing for photos next to a cutout of Sir Hillary. I have never understood why serious rock climbing friends get so upset whenever I have mentioned this idea over the years.
In selling away the rocket equation you buy yourself drag. Lots of energy up front means lots of drag. Overcoming that drag means more energy and, hence, more drag. The middle child of this solution is tremendous G force.
You can't slingshot complex things into space. Pretty much just fuel and raw materials. That means you need in-space (a) refueling and (b) additive manufacturing capabilities. There are teams working on both problems, though they are presently in the domain of science versus engineering. Perhaps that will now be incentivized to change.
[1] https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation