The game is specific in calling it a cathedral project, which means a cathedra is already attached and (presumably) a Bishop also. Why Halifax is building and paying for the project when the local Church alone probably earns and knows more than he does, is a little unclear. Paying or funding a monastery was not uncommon for a ruler (which is why I'm unsure why that isn't the option rather than the cathedral) but an actual cathedral is more like a royal or imperial project usually (if only because it takes decades at best to build. York Minster took 400 years).
This all being said, I suspect you are right and this actually is a Minster that no one actually wanted but Halifax decided to build anyway (which is why he is in charge of everything). Its impressive he did it all in 25 years, to be honest. At some point, either it'll burn down or someone will actually care enough to finish the damn thing properly.