A new head into the 9900 is on order. Thanks for the replies. Will let you know how things go.
Installed the new head and have been printing for about a month. Definitely seems like the right decision. The Epson-certified technician who installed the head for me said as much. In his experience, the 9570 was an advance in printer technology that was not yet fully functional. Evidently the last few years have brought him many service calls.
Though I did install my own new ink selector assembly recently, installing a print head seems much more involved. It's not just a question of getting it in place, but of testing and calibrating. Calling in the service tech was definitely worth it.
I expect to go on for two or three more years with the 9900 with a new head, scrupulously printing every day or two, shaking carts, running nozzle checks, keeping constant 50% humidity in the studio, and performing all the other rituals that keep the printer printing. Right now the results are as precise, bright, and detailed as ever. I'm not sure the 9570 would have been an improvement, despite the wider gamut.
This was my second 9900, 7 years on the same head. Meanwhile, I have my first 9900 in the studio with a cyan head that doesn't print. I have thought of converting it to third party inks, particularly Jon Cone's piezography inks. A printer I talked with in Florida was converting 9900s to dye sub and printing heat transfer images for t-shirts. All very interesting, but I probably just need one printer.
Thanks again for the good advice. It really helped me to make a decision.