I'm sorry, I'm not entirely sure what you mean. All my photos are on my local hard drive on my computer at home, and I just point immich at that folder.
Once it boots up, immich doesn't need internet to function, except for the first time you use image processing (the machine learning to search for things inside of images) and geocoding (putting the images on a map). Once it downloads those for the first time immich can run completely offline on your local computer. If you wish, you can open immich so you can connect to it from another device, such as a phone, but your images/data never leaves your device
That's fair. And in the case of Immich, the photos are automatically backed up to my computer so if I lose my phone I don't lose the images, but I totally understand wanting something exclusively on the phone (Immich can let you browse photos local on your phone, but I don't think it indexes them for searching by person/object). Unfortunately I'm not sure something like that exists, especially since the machine learning for identifying people/objects in pictures is pretty computationally intensive.
I wish you luck in your search! I agree that would be a great app to have