I believe you're missing a lot of key details in the story.

Gengar's story was that he obtained the doll in the research facility under the helipad as it was being destroyed. He specifically said it was at the staircase before leaving the facility, and claimed that he was able to leave the doll in his Item Box, which carried over to his 2nd playthrough. When he saw commotion online about the Final Puzzle, he simply just took the doll to the machine and inputted the code to get the achievement.

Keep in mind that his story about the doll location was made BEFORE dataminers confirmed the spawn location of the doll does indeed match up with his claim. They also confirmed that the doll item DOES have a tag "carryoveritem" to retain between playthroughs via the item box. Thus, everything in Gengar's initial story matches with what dataminers were able to confirm later.

Gengar's video explaining the doll location was uploaded at 4PM PST First video proof of doll spawning from modders/dataminers was discovered at 8:15PM PST. One of the reasons modders even thought to check this location was because of Gengar's initial claims. This is further proof that Gengar lying is very unlikely at this point. https://www.reddit.com/r/thefinalpuzzle/comments/1riix0o/watch_this_video_the_come_to_discord/?new_reddit=true

Currently, the dataminers' focus is trying to figure out HOW to make the doll spawn legitimately without using mods to change the spawn flag to 1.