Can I use an old chimney as a electrical chase

TLDR:

Are there any code/cost reasons we couldn’t run BX/romex to our attic through an old disconnected chimney?

Long Question:

I will be closing on a house shortly and there is a hole in the attic floor where the house’s original chimney used to run (the chimney has been removed down to the attic floor). The attic is finished but run-down and we plan to eventually renovate it; but there is limited electrical. Before we move all of our furniture in we were going to have a contractor refinish the floors and replace any soft spots or holes, but this seems like a perfect chase for electrical and I was considering bringing in an electrician to run an extra circuit to the attic before we cover it up. Would there be a code problem running through an old chimney like this? Or alternatively would there be another reason this would be worse than trying to fish a cable up past two stories of nice old lath and plaster walls?