Re: Sorry Mr. Flaxman, but it is MY printer, not yours....
Unpopular Opinion Alert:
Yes, you own the printer: A disposable piece of plastic, copper, and whatever they scraped off the hooves of dairy cows who've stood in wet pastures. It has little actual value, and even less if it doesn't print.
They can design it however they want, up to and including encoding that only accepts their refills. You may not like it, but HP has not a secret of their stance on refilling HP cartridges and third-party cartridges. Their manuals and documentation are quite clear that you are to use an HP replacement. Using a cheaper solution is a risk you take; it's unsupported and there is no promise by HP that it will work in the future. When they do disable it, all those previously purchased black ink cartridges from Bob's Discount HP Shoppe will become useless.
If HP was the only printer game in town, I might be more sympathetic to your plight; without other options, HP would be abusing their market position. But there are numerous other options (named earlier in this thread) that can do the job, and maybe even better. That you choose to stick with HP is an issue you'll have to come to terms with.