Somebody please confirm, but my understanding is that the legal minefield of existing patents is vast, really hard to navigate and easy to sink your project.

Patents aren't granted forever. They expire after 20 yrs. The real barrier isn't the printer, but the cartridges and those designs keep changing. So if you want to have a successful open source printer design it needs to have a ready source of open ink cartridges as well.

I honestly hope this will happen in our future. But we're in the era of extortion capitalism with walled gardens and captive customers.

I think it doesn't exist because the Brother brand is already good enough

For inkjet technology, it's the print head. Fundamentally, an inkjet printer works similar to a plotter, laser engraver, FDM 3D printer, or CNC router: There's a mechanism that accurately places the tool head at the desired X and Y position in order to do its business [0]. This part is a solved problem, there are many open-source reference designs to draw from.

For a printer, the devil is in the details though - quite literally: A printer needs to place tiny ink dots on the work piece in a fast, accurate, predictable, and repeatable manner. Placing just one single dot at a time would be prohibitively slow, so the print head needs an array of tiny nozzles in order to print multiple rows in one pass. The design also needs to ensure a consistent (and adjustable) dot size across all nozzles regardless of the ink level.

Designing and manufacturing such a print head is extremely complicated (if not impossible) for a hobbyist, and as of today there is no vendor-agnostic third-party design you can just buy off the shelf. The only forays into DIY printers are either rudimentary low-res single-nozzle designs [1], fully repurposing existing printer hardware [2], or reverse-engineering proprietary ink cartridges that come with a built-in print head [3].

[0] Technically, this is not entirely correct because a typical printer's Y axis is a rotating drum that transports the paper instead of a linear axis. Also, a printer's X axis is usually equipped with an encoder ribbon that allows for far more accurate positioning compared to the classic open-loop stepper motor systems used in many open-source designs.

[1] Example: https://hackaday.io/project/167446-diy-inkjet-printer

[2] Example: https://www.youtube.com/watch?v=_VP8ORq2al8

[3] Example: https://ytec3d.com/hp45-inkjet-printhead/