Is autofocus adjustment actually no longer required on phase detection sensors that are placed on the image sensor? If not, why? How does lens autofocus adjustment come into play?
It does not depend on the sensor type (dedicated PDAF vs image sensor PDAF).
The need for adjustment comes from the fact that the phase-based autofocus works by using sensor data to determine the absolute lens focusing distance. If what the lens does is not what the camera commands (because of the physical imperfections, incompatibilities across lenses) the focusing precision suffers.
(By contrast, the contrast-based autofocus ;-) drives the lens without knowing or needing absolute positions - and therefore without a need for precise adjustments. It just tells the lens to move forwards or backwards, until the result is acceptable. Even if the lens machinery is slightly off, it will still eventually come to the correct focus.)
On a deeper level, no digital camera really needs autofocus adjustments. It is a relic from ancient times, when film cameras could not verify what is really captured. Nowadays every camera might simply learn from the immediately accesible results and auto tune the phase detection autofocus distances with little user intervention. But as far I know, they don't do it.
Consider the evolution of AF systems over the past several decades:
With traditional DSLR cameras, fine tuning of PDAF autofocus based on a separate sensor is needed to overcome several different possible errors. Among those errors are:
With most cameras that use focusing based on information from the imaging sensor, none of these things really matter since the final focusing is done using contrast detection, rather than phase detection, methods. The earliest CDAF systems had no PDAF capability so all focusing was done using CDAF. But this is also the case for cameras that have "hybrid" sensor-based PDAF combined with CDAF.
A simple analogy between focusing using a dedicated PDAF sensor and focusing using imaging sensor based CDAF (including hybrid PDAF+CDAF) would be to compare focusing the lens on an old Speed graphic camera to focusing using an old manual focus SLR.
With the main imaging sensor based PDAF-only system used by Canon's Dual Pixel CMOS AF in Live View for both video and still imaging, aspects of off-sensor PDAF and on-sensor CDAF are combined. Instead of a one-and-done AF measurement followed by a single instruction to the lens regarding how far to move, which is what typical PDAF does using a separate PDAF sensor¹, Canon's DP CMOS AF takes additional measurements as the lens is being moved, which is what is normally done with CDAF. This is possible because PDAF is being done using the main imaging sensor, thus there's no time used to flip up the reflex mirror between the PDAF measurement and before the image can be captured. PDAF can remain active until the imaging sensor needs to be cleared in advance of capturing the image without slowing down the camera's shooting speed.
There's obviously no difference in the length of the optical path from the lens to the sensor since it is the exact same sensor.
Adjustments to the lens position using 'rinse-and-repeat' measurements are also not as susceptible to errors in the amount of movement instructed by the camera compared to the amount of movement actually made by the lens, because the camera continues to optically measure focus and update it instructions for how far to move the lens until it finds the position that has the highest amount of contrast², rather than telling the lens how far to move one time and then depending on the lens to precisely follow that instruction without optically rechecking.
I infer on-sensor pdaf without cdaf still has to overcome errors (i.e. flange focal distance, lens positioning). Does on-sensor pdaf allow to overcome such errors maybe by iterations?
Most, if not all, of the errors are the same for focussing as for when the image is actually taken, and are thus cancelled out, because PDAF is being done in the main imaging sensor instead of a separate PDAF sensor. Main imaging sensor based PDAF also has the advantage of not being 'blind' without a reflex mirror blocking the main imaging sensor. There's not near as much of a speed penalty for remeasuring focus after the lens is moved following the initial measurement and before the image is taken without a mirror in the mix. By being able to take incremental measurements as the lens is moved, errors resulting from differences between intructed and actual amounts of movement can be significantly reduced. A 1% error for a 0.1mm movement will be significantly less than a 1% error for a 10mm movement by a factor of 100.
¹ Even in 'Servo' AF or 'Continuous' AF, the process is One PDAF measurement → One lens movement → One PDAF measurement → One lens movement → ad infinitum until the shutter button is fully pressed. There's no refinement of the lens position to "zero in" on a stationary subject distance. Rather, it's based on the assumption the subject will most likely be at a different distance for each PDAF measurement → Lens movement cycle. Each additional measurement is done to measure how the subject has moved, not how close the lens got to properly focussing the subject on the previous measure → move cycle. Predictive logic is also used to estimate where the subject will be in an x.x number of microseconds when the shutter opens after the mirror is flipped up.
² Both PDAF and CDAF depnd upon contrast between details in the subject to measure how focused the lens is.