Not necessarily. I'm not a rocket scientist or economist, but it is possible that FH is less cost-effective per kg on LEO than F9. You need to accelerate three boosters to several thousand km/hour and then to decelerate them to 0 km/hour at ground level. And then to do all the maintaining on those boosters, which takes about a month. Does it allow for 3x payload?
There are other potential issues, like a volume: a payload is not just mass, but a volume too. It needs to fit into fairing. Adding two more boosters doesn't increase available volume.
> I still believe it'll take quite awhile to match the economics of an F9 launch
Musk says that he relies on Starship launches to make it work reliably as expected. And yes, I believe that it will take years. Maybe it will be faster than with F9, but not an order of magnitude faster.
Mass to LEO (from the SpaceX site): F9: 22,800 kg; FH: 63,800 kg.
So 2.8 times. Both expend the second stage, so if they recover all the boosters then FH possibly has lower cost per kg, but they're not saying.
These deployers can cost $10000, or even more depending on the specific cubesat size. So when Starship allows small payload aggregators like Spaceflight Inc to recover the PPOD/deployment hardware then it’s going to provide even further cost reductions beyond the simple dollars per unit mass lifted to orbit. The reusable nature of Starship will be a big part of how it changes the economics of space related industries.
Falcon Heavy is again evidence of this. We're not seeing ride-sharing on FH. We're seeing large (primarily military) payloads that F9 just can't do.