To be fair, I expect there is also an element of him wanting to get a bigger payday if he leaves on a free, so he might have rejected a few offers himself this summer.
But regardless, we shouldn’t be extending him imo. For argument’s sake, if we can hypothetically find somebody with 75% of Wilshere’s ability but who makes 45+ appearances in the season, that hypothetical player will have contributed far more to us than Jack. A reliance on Wilshere to be a squad/first team player is risky, because if he does get injured then you start having to rely on youngsters who haven’t broken into the team yet, or playing people woefully out of position.
The fact that Jack takes up a squad spot and a salary of c. £4m - £7m per year is the reason I think we shouldn’t keep him. That squad spot and salary could be more economically/efficiently spent elsewhere i.e. on a player who is going to play double or triple the minutes that Jack will.