Assuming the cartridges work, go for it, but you might run into DRM/issues using third-party ink.
I guess converse should be possible though - if I put in regular commercially bought carts, these shouldn't come off my print total.
But DRM on ink? What's the worst that could happen? I ruin the printer I paid for? I ruin the page I'm printing? What's the justification, outside of blind greed?
You did mention the real reason though.
1) Counterfeit cartridges may cause damage to your HP LaserJet printer hardware – which can lead to downtimes of the whole company’s printing infrastructure and disruption of entire workflows.
2) Manufacturing and distributing counterfeit items is illegal. So, buying fakes may bring you and your organization into contact with shady business which may be operated by large-scale criminal syndicates.
3) Counterfeit toner cartridges may alter the printing behavior of your HP printer. When using Original HP toner cartridges, only then can HP printing systems perform within Blue Angel guidelines limits.
4) Due to their often poor print quality and necessary re-prints, counterfeits are likely to waste valuable resources such as paper or energy. Also at the end of life, counterfeits likely don’t get recycled.
Directly copy-pasted from https://www8.hp.com/us/en/cartridge/anti-counterfeit.html
Whatever the reality, they've made it so consumers get burned by counterfeits (rendered completely nonfunctional) instead of HP, while conveniently eliminating legitimate competition.
Also, with different ink your printer might print content with a different color profile, so it won't look the same, and it might need to clean the printing heads more often (and the cleaning tech for that is often only designed for a limited amount of uses).
But while all these are legitimate reasons, projects like Instant Ink show the true reason is to just turn the printer into a subscription product: buying a printer at a loss, and making a profit with every page printed.
I think argument from HP is pure greed. I don't know if these actions could bleed into the laptops as well. I currently have a HP, would have to look at Dell or something else in the future.