You are, by the very definition of using your reaction to take your action via Ready, is on
another creature's turn. It isn't
your turn. You can only take part of your turn--not the entire thing. Your moment has passed by and on
your turn you took the Ready action, otherwise when did you take it??
So, you aren't taking the Attack action on
your turn, which is a requirement for Extra Attack, you are taking the Ready action
to attack, so you only get one attack.
Again, you can certainly
play it however you want, but one piece of SA, which is only a general piece of advice instead of specifically talking about Ready and Extra Attack, against every other instance I can find isn't going to convince me, especially when the piece you have says you only take part of your turn.