I'll try a little bit of Fermi estimation.
Mass Limit
Similar to rockets, there is an exponential equation for tethers. The Wikipedia article on it is actually quite solid in terms of the gritty technical details. Easily enough, we can plug in some velocity at the point of release and use the material characteristic velocity for M5 fiber. As far as I can tell, this is the most reasonably optimistic out of the practically available materials. Then the ratio of the mass of the tether to the mass of the rocket will be given by:
$$ \frac M m = \sqrt { \pi } V_r \mathrm{e}^{ {V_r}^2 } \mathrm{erf} ( {V_r} ) $$
Keeping with the spirit of Fermi estimation, let's image a Falcon 9 with a launchpad mass of 500,000 kg, which puts 10,000 kg into orbit. If we use an engineering factor of 3, then we can easily calculate the velocity at which the catapult assist system will weigh as much as the rocket on the launchpad. This is in the neighborhood of 3 km/s.
It's important to not go much further than this mass limitation. No company does more than a few launches per year. If per-mass the tether costs a similar order of magnitude as the rocket, then the mass ratio is the number of launches you'll need to amortize the cost. If the tether material is 10% the cost of the entire launch assist system, and you need to pay off investors within one year, then we're clearly settling on (mass of rocket) = (mass of tether). It doesn't matter much if you change your financial risk tolerance here, because the mass equation is so insanely exponential.
Needless to say, this could still work. If a rocket reduced its Delta V budget by 3 km/s, then it could practically double or triple its payload delivered to orbit. You would be tempted to think this makes sense. Actually, in space, it does make sense. The problem is that we're still on Earth.
Length Limit
Strange enough, the above mass ratio equation says nothing about the tether length. It could be long or it could be short. If the edge velocity is the same, it doesn't change the total tether mass. If you go faster, the centripetal acceleration is higher so the tether must be thicker - and this perfectly balances the effect of a shorter tether.
Thus, g-forces actually limit the length of the tether, and these are constrained by biology. Let's say that our limit is 3g. The equation is simply $3g=\frac{v^2}{r}$. If you plug in 3 km/s, you get a tether radius of 300 km. Oh my. This will not do.
The most radical version of the tether launch assist system consists of either high altitude balloons or aircraft. These are limited in altitude due to atmosphere density, typically to about 10 km in the case of large jets. Combining the limitation of 3g and 10 km, you find that the launch assist system can only aid to the tune of 0.5 km/s. So with this system, you can only improve the rocket's mass fraction by about 20%, but only at the cost of reducing the starting mass by the requirement that a jet or balloon can pick it up in the first place. More than likely, it hurts more than it helps!
Ways Around Length Limit?
Your idea was to reduce the propellant the rocket needs. You can see from my calculations that the limitation isn't the bulk mass of the tether. The problem is that the length need is so high that you don't have anywhere to attach it to. But your idea of attaching it is:
Imagine same device attached to a space rocket horizontally.
Sounds like the tether connects the rocket to an anchor (I assume a moving anchor) on the ground. But the problem is that the rocket must pull the tether up with it here. That increases the amount of propellant the rocket needs, which is already violating your objective.
In fact, this is why any serious proposals in this family involve launching hardened commodity payloads, not people, and not conventional rockets. By increasing the acceleration, you can keep the structure to within roughly a kilometer scale. But such a megastructure is a hard sell when it will only marginally increase the payload in the first place.
Alternatively, you could try a scheme to "attach" it to something not bound by terrestrial constraints. For instance, something already in orbit. This is the approach of various partial space elevators. These can reasonably be >300 km in radius, but then the mass constraint pops back up again since you must put it in orbit to begin with.