I agree that the iPhone app only has the opening explorer database. I don't see the equivalent to the link you cited.
Exploring Openings in the App
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The app seems to only have the opening explorer (i.e. where you place pieces on the board and chess.com tells you what the position is), but does it have the openings page where it gives you the most common openings using E4 and E5, etc. (linked here: https://www.chess.com/openings)? I can't seem to find it.
Using an iPhone.