My loose understanding of the story was JJP got the POTC and Toy Story license around the same time (when the TS4 movie would have been in advance stages of pre-production). People did not like the lack of film clips in POTC. JJP went back to re-negotiate the Toy Story license to get movie clips....things happened and years went by. Toy Story 4 with movie clips was the result
I also imagine Toy Story 4 with movie clips was far cheaper than Toy Story 1-3 with clips.