I have used third-party ink carts for both Eason and Canon printers and black only carts for several different Brother lasers. Only problem I ever has was with the Eason ink jets which had trouble with drying up print heads. I have only ever used third-party inks in the Canon and have never had any problems with it. Likewise, never had problems with any Brothers using third-party toner.
I have absolutely no experience with other printers.
I do not seek the cheapest ink/toner. I do look at ratings of the company, its customer satisfaction and it’s time in business. The price of ink/per gram for most OEM carts is close to the price/gram of gold. Toner prices are not much better.
Printer manufacturers are just following the example of Gillette. They have traded traded razor blades for ink/toner.