There are legitimate advantages to high altitude launch. A big one, perhaps the biggest, is that exhaust pressure of a rocket needs to be at least as high as the ambient atmospheric pressure at launch. This limits the expansion of exhaust and the sizes of nozzles for rockets that are launched at sea level. A notable example is the Falcon 9 launcher, which has 9 engines on the first stage and one similar engine with an engine bell that is almost the same diameter as the entire rocket for the second stage which only operates in a near-vacuum environment. Higher expansion translates to higher exhaust velocity (Isp) for identical propellants. Since the rocket equation is exponential with respect to the ratio of delta-V and exhaust velocity, even slightly higher exhaust velocity can translate to significant increases in overall stage performance.

To use the example of the Falcon 9 again, consider the second stage with a dry mass of 4 tonnes, 107.5 tonnes of propellant, and an exhaust velocity of 3.41 km/s. With a 5 tonne payload the stage can achieve a delta-V of: ln((107.5 + 4 + 5)/(4 + 5)) * 3.41 km/s = 8.7 km/s. Now, if we substitute the first stage sea level exhaust velocity of just 2.76 km/s the overall delta-V is 7.1 km/s, and to get to the same delta-V (8.7 km/s) now requires a mass ratio of 23.4:1, resulting in an effective payload of just 0.8 tonnes, less than 1/6th as much as the original payload!

In actuality the overall performance differences won't be quite as severe but this does point out the potential for a rocket launched from higher altitude to have equivalent performance to a much larger rocket launched from sea level.

Another major advantage is operational flexibility. You can launch from a wider variety of launch locations, which might make it easier to launch in a more timely manner, or could make it easier to launch certain kinds of payloads (which don't need to be shipped around the world, the launcher can come to you). And potentially it makes it possible to launch from more equatorial locations, gaining a slight orbital mechanics advantage.

However, all of this comes at some pretty steep costs. With this design it's harder to make incremental changes to the vehicle. One of the classic mechanisms for improving launcher performance is stretching stages, but that's out of the question here. You can't make the rocket bigger without making the carrier aircraft bigger as well.

Additionally, and most damning, the tradeoff here is a bad one. On the one hand you have a sea level launched rocket which needs to expend more propellant to reach orbit. However, propellant is dirt cheap, it's the cheapest part of a launch pretty much. Especially for a LOX/Kerosene rocket the propellant isn't even going to crack 7 digits in terms of cost, even for a big rocket. With high altitude launch some dirt cheap propellant is saved at the cost of an inordinately expensive carrier aircraft, and a very complex and dangerous flight profile (you need to have a piloted aircraft takeoff with a fully fueled rocket). Moreover, you've now cut off all of the most straightforward avenues for incremental improvements of the launch system: you can't make any part of it bigger without starting from square one with a new iteration.

And in a world where reusable rockets are becoming ever more commonplace this makes even less sense.