I'm trying to solve this problem where I find Vout/Vin if rπ if infinite. I also have to conside early effect. When I try to solve it, I get Vπ = 0 and that makes gm zero, a bit confused on that.
I don't care who is explaining gm in transistors circuits because it is incorrect. That is why you are stumped.
Common base circuit, is voltage amplifier that has the current gain (approx) as 1 but is a calculation of Alpha Trans-resistance Factor (instead of Beta) is explained:
Of course you looking for voltage gain, it has no trans conductance because its a factor of a voltage drop across the emitter and collector resistors in parallel when you look at the AC analysis equivalent.
Of course, the gain formula's expression:
Images from: Electronics Tutorials - Common Base Amplifier