for all their talk of wanting privacy, it'll be too good of a PR opportunity for H&M to miss
I don't remember them saying anything about privacy, rather that Archie would be bought up as a private citizen... Which means something else. This wording is always used against them.
This has been reflected in them not taking up Earl as a title for Archie, which he was entitled to (Earl of Dumbarton). Also, they said they they would have the christening at a private chapel (Prince George and Prince Louis were christened at the Chapel Royal and Princess Charlotte had a paddock open to the public) with a photo opportunity and this is what they did.
H&M never said that they would not take Archie to events or never release photographs of him. As Archie is not the son of a heir, he will obviously not be in the public eye as much as G, C and L.