On flat bars bike (mountiain bikes, hybrids), the integrated ones don't make much sense to me, and have precious little difference, ergonomically speaking, from separate controls. And, as OP said, complicate things if you want to change a component. There is also benefit to being able to adjust the position on the bars of one set of controls relative to the other.
On drop bar bikes - road, touring, CX, ect - the riding position and bar shape makes brake and shift levers in separate places a challenge. Older solutions include shift levers mounted on the downtube of the frame and on the ends of the bars, and the integration of shift levers into the brake levers (c. 1992 or '93) was a major step forward for bike ergonomics. I still use bar end shifters, but that is because that is what I have, and I am too cheap to upgrade.