The altitude data on ATC screens comes from the static system of the aircraft, because that is the only sufficiently precise source. The area primary radar is, at long range, only accurate to maybe a mile and that applies to vertical position too. For horizontal position when the separation is 5 nautical miles this is good enough. But the vertical separation is 1,000 ft and the radar simply can't distinguish that. So the aircraft transmit their altitude in the Mode C and Mode S (ADS-B) and that is used for altitude display.
Also, the on-board instruments measure barometric altitude and that is what aircraft are instructed to fly. But a precise radar can only measure geometric altitude, which due to changing atmospheric condition is different, so it wouldn't help anyway.
In future, the position data will more and more be also obtained from the aircraft. The ADS-B broadcasts GPS position and this is more precise than the primary radar and is available even over open sea where primary radar coverage is not possible. Therefore newer systems depend on what the aircraft itself reports more, not less.