For a lot of his time at Barca, Sanchez was really good. A Barca forward can only be good to a certain level because the team is based on Messi, and within these limits Sanchez was usually fine (with some significant periods not good).
Barca ended up last season not a very good team. It is quite reasonable for the team to significantly overhaul now, even send off some of their better players. Especially a player who gets a huge selling price like 42M euros.
Sanchez has been better for Chile than he ever was for Barca. He may do very well with Arsenal, he will certainly have more opportunity there.
The team was not lying that they really liked Sanchez. There is no way he would have been sold unless the team thought they were getting a more talented replacement.
Suarez was the best player in the EPL last year, even with the suspension. Barca is rolling the dice for a big, big talent. Neymar, Messi, Suarez could become the best front line ever.