1. In round numbers:
. ~9.5 km/sec to LEO (given, approximate)
. ~6.0 km/sec from gas gun (FTA)
. ~0.5 km/sec atmospheric drag (FTA)
= ~4.0 km/sec needed from projectile rocket
. 350s ISP for projectile rocket (assumed, optimistic)
= 0.69 propellant fraction
. 450Kg projectile (FTA)
= 310Kg projectile rocket propellant
= 140Kg projectile non-propellant
. ???Kg projectile structure, motor, etc.
= ???Kg net cargo to LEO (in any case, 140Kg)
2. Assuming you want to rondezvous with something in an established orbit (e.g., the ISS), any significant orbital maneuvering is out of the question; in paticular an orbital plane change--whether by the projectile or the target--as it's too expensive.
That limits the number of launch windows. You can't simply launch projectiles into orbit as fast as the gun can fire, otherwise you'll end up with them scattered in various orbits that you have to chase down (again, very expensive).
E.g., there are nominally 2 launch windows/day for Shuttle flights from KSC to the ISS. (Due to various rules, in practice it's limited to 1/day, but we'll ignore that.)
3. Even with optimal launch parameters, orbital rondezvous is still non-trivial, and one reason why even unmanned ISS resuplly vehicles are much more than simply a dumb ballistic container, and have, e.g., OMS and RCS motors, propellant and the weight/complexity/cost penalties that come with them.
Which is why larger, more infrequent and expensive missions will remain the norm for the foreseeable future--with or without a space gun or its ilk.
4. In short, we need an orbital infrastructure that can handle smaller/dumber vehicles. That doesn't exist, and few if any of these proposals account for it. With, e.g., a group of ion/electric tugs it may make more sense. That is, something that can cost-effectively collect those smaller/dumber vehicles and bring them to where they're needed.