It would be a lot more aspirational to figure out how to build a tank-based printer (although maybe its not possible to compete with the incumbents on price in this space)
In theory you could probably modify one of these by grabbing an HP cartridge and slapping an ink tank on top with the right sponge for transferring the ink over. Selling parts from HP printer cartridges might creep a little too close to HP's patent lawyers, though, and having users open cartridges to extract the print heads themselves sounds like a messy solution that would reduce the appeal.
I'm sure that given time, someone will 3D print an ink tank holder that will let you use cheap ink supplies through the expensive HP cartridges.
Yes, but it also means it won't be cheaper or more reliable than a regular printer. The print head is what fails most often and you'll still have to buy a new one.