1. Page One's mission was to kill Sanji = Sanji is alive, thus P1 didn't win.
2. Drake had a mission of finding and killing Sanji, Hawkins was trying to find Law
yet they both showed up at Yasue's execution to control the crowd
which means their missions got put on hold and were given this new mission
yet Page One was nowhere in sight. He would've been with Drake at execution.
3. Sanji said he will take out P1 and P1 was not seen for like ~50chapters (10days in manga)
Which goes in line with what Sanji said. P1 was nowhere in sight to interfere with alliance for those 10days.
4.
"B-But Page 1 showed up in Onigashima during the raid that means Sanji didn't beat him"
Page one had 10days to recover
No one in One Piece can take out someone for 10days. Even Yonko couldn't damage Page One without him recovering in 10days (except killing P1).
Kaido took out Luffy= Luffy recovered the next day
Zolo and Killer took each other out= Zolo recovered the same day. Killer was shown fine the next day, ready for torture.
I doubt Sanji's
3 normal kicks can put down an ancient zoan user for
10days
That unironically would be the best feat in One Piece if it was the case.
And I doubt Sanji beat Drake offscreen.
Last time we saw him with drake, he was dodging him while looking for Hiyori
I doubt Sanji was fighting seriously in order to beat Drake. He probably escaped with the rest SH.