I had, in the past, also been thinking a bit about how the system could work offline because it would avoid a lot of security issues for something like a medical office that wants to create content that's then copied into a medical records system.
My initial thought was to bundle a local copy of the server with pre-trained models, but that becomes problematic on mobile clients. I'm writing the server in Go, so if I go that route I'd probably need to reimplement parts of it in another language and avoid using any remote APIs.
So the answer is: very likely yes, but not initially. You've definitely moved the functionality up my planned features list.
I always prefer a Free software/open source solution - but I'm not sure how you'd monetize that. Maybe charge for the app (ios/android) - and provide a free/open self-host server solution, along with a subscription service and a web client? (The payment for the app would also grant access to the subscription, and for those that didn't want to self-host/wanted to support the project could pay for the subscription and use the web SaaS solution?).
I've been an OSS user and supporter for a long time. I don't think I'll be open sourcing the core system anytime soon, but I'm very likely to release any useful NLP or ML-related libraries that are created as part of this project. Not sure what those will be or when they'll be ready, but I do have a mind to give back to the OSS community.
Not sure what kind of income they see, though.