There is a big difference between the output swing and the input swing. By checking with unity gain connection, BOTH are being evaluated.
In other words, even if the output can swing rail-to-rail, the output may be limited by the input common-mode range in follower connection.
The output swing is limited in a telescopic OA by where the devices leave saturation and go into triode, as the original poster said.
Check it in simulation by sweeping the input voltage and looking at the output voltage range. For low distortion, output must be well within these limits.