Your question seems to be about the purpose of overlapping during synthesis in addition to overlapping during analysis.
In analysis, as perhaps you know, the overlap allows better capturing of temporal effects that span consecutive analysis blocks.
For instance, rapid transitions in low pass systems have long time domain responses, and how that response splits between subsequent analysis blocks affects how each block represents it spectrally.
If the transformed signal is not processed, and the synthesis is performed by simply inverting each block individually and stitching the blocks back together, then the time domain signal will be perfectly recovered and perfectly represent the original (quantization aside). This is regardless any effects that may span multiple blocks, or how the blocks align with effects in the signal.
However, if the transformed signal is processed in the frequency domain, and independently on consecutive blocks, then the effect of a rapid transition on the processing is different depending on where in the block this transition happens.
But it shouldn't be: an artifact in the signal (in the sound) should be identified, extracted and processed regardless how the artifact lines up with the block timing, right?
To somewhat ameliorate these unwanted block effects during analysis, blocks are overlapped and windowed.
The same goes for synthesis. When processing a signal block-by-block in a transformed domain (e.g. in the frequency domain) the result of the detection and transform in one block may be different from the result in another block.
And if, over time, there is a block-by-block change in the processing (e.g. a gain or a tuning parameter is adapted based on a frequency analysis), we would want this adjustment to be gradual in the output, and not audible/discernible at block boundaries.
To prevent discontinuities at block boundaries, after processing, you would overlap the blocks at synthesis.
If you listen carefully to "auto-tuned" singers with poor pitch, you can sometimes hear the block based chopping of the sound. This is usually prevented with sufficient overlap during analysis (to detect and correct) and during synthesis (after re-tuning).
The same goes for speeding-up video without raising the audio pitch. It is the overlap during synthesis that prevents block based chopping. Usually it works well, but for too much speed up you might find that the chosen overlap is not sufficient to prevent these artifacts from being noticed.
(You can observe this out on youtube with 1.75x playback speed and classical music with drawn out notes)