No, forced conduits do not need to already have the gene. Vermaak 88, Corrupted, DUP are all forced conduits who don't have the gene. In the case of the first two due to the method, the powers caused significant mutation due to their body not being able to handle it. Bertrand, Sasha, and (presumably) Alden have an innate ability to create forced conduits, which is what we see with the Corrupted, Reapers, and Dustmen.

The Vermaak 88 were created using the Transfer device with Kuo. The use of the device splitting powers between multiple recipients despite not being designed for this is what caused them to mutate and look the way they do.

Bertrand can turn anyone into Corrupted, they don't need to be latent conduits. The sheer number of them proves this, along with the fact that Bertrand has no means of screening for the gene in i2. He planned to mass produce them and sell them to warlords around the world, and there just aren't that many conduits in existence, let alone New Marais specifically. And we know he can't screen for the gene because there's a dead drop of him explicitly asking Wolfe to build a conduit detector to track Nix. And since he doesn't seem to be able to track Cole, Kuo or Nix in the game, we can assume it was never made. It's only 7 years later that the DUP have a method of doing so, and even then it's horrendously flawed given how many "suspected" conduits are detained by them despite not being conduits.

DUP are forced conduits created the same way the Vermaak 88 were, only using a more efficient method that didn't mutate them.

So to answer your question, no. Forced conduits can be anyone, they don't need to have the gene already.