The reasons I give are perhaps not all the reasons but they are the most important ones. The reason that Vieira wanted to leave is of course valid, but I believe that had Arsenal and Wenger in particular really wanted him to stay, he would have.
Reason 1: His form has not been particularly good. Even during the unbeaten season he was not at his best. And few Gooners would call last season anything but his weakest since his first 15 months at the club. It's not as if he was terrible. From time to time he showed some brilliance. But in most games he seemed to lack interest and passion. Most of the time he has seemed as if his mind has been elsewhere.
Reason 2: He is not a Captain Fantastic. Obviously no player could carry the captaincy to the same height as Tony Adams did, so one shouldn't criticize him in that respect. But what one could require, and that I believe Wenger thought Vieira would develop into, is that the Arsenal captain is a strong leader on and off the pitch. If not vocal, he should at least lead by example. Last season this was not the case. Furthermore, the board could hardly have felt it appropriate that their team captain was encouraging Cole to make threats in the negotiations for a new contract. Was the team captain acting for the good of the club or for the good of his friends?
Reason 3: 22 million pounds. 2 seasons' wages and the 13-14 million from Juventus. It is very likely that Vieira was not going to sign an extension to his existing contract, and would leave Arsenal on a free in two years.
Reason 4 (the most important reason): Patrick Vieira is a central midfielder who does not form partnerships. Vieira plays his game and needs someone to fill in the spaces that are left over. Petit, Grimandi and Vivas were all defenders moved into midfield for the purpose of covering for Vieira. The central midfield aspirations of Stephen Hughes and Ray Parlour were severely damaged by the fact that they were not free to play their natural when playing with Vieira. A measure of this is that Parlour's best games in the middle were played with Grimandi or the later arrival Edu.
In terms of results, Edu and Vieira had success together. But Edu sought partnerships in the middle, which so splendidly was proven together with Parlour in the game at the San Siro. Our best result (and performance) in Europe, achieved without Vieira. The flow of football that that Arsenal team showed has never been repeated in European football. Instead the Vieira dominated Arsenal midfield has been unable to find space, and nothing has been created and very little has been achieved.
The arrival of Gilberto Silva meant that Vieira's ideal assistant had come to join him in the middle. It also meant that Edu would never be first choice again. But this just increased the unlikliness that an Arsenal team would gain real success in Europe. Tony Adams wrote in his biography that Arsenal would never win the league championship with Ian Wright as striker despite his outstanding skills. The same can be said of Vieira, Arsenal could never be the Champions of Europe playing the Vieira-midfield.
With the gathering of strengths occuring in English football at the present, even the domestic near future for Arsenal may have meant more dropped points at bottom clubs rather than the glory of trophies. The coming season is all about challenging the best teams with Arsenal's best TEAM. Despite winning the FA Cup, that performance underlined that the problems that are so apparent when playing European football were appearing with increasing frequency in Arsenal's domestic game as well.
At Arsenal there is perhaps the most promising central midfielder in the history of the club. Wenger has made the choice not to suffocate the talents of Fabregas by forcing him to play as a utility defender in the middle. In the choice between change and no change, the results and performances of the last few years have swung the decision made by Wenger in favor of Fabregas and the possibility of international success. The recent purchase of Hleb points in the direction of a change in the way Arsenal play their football. A so-called number 10 is not really the kind of player we have had before. The same can be said of Wenger's target Baptist. We have never played with a big man up front who can win the ball in the air.
This change in the way Arsenal play football may also explain the apparent hesitation in resigning Bergkamp and Pires. (At their ages) can they adjust? The basic formation will still be 4-4-2, but without Vieira there will be changes in the movement of the central midfielders. Wenger has pleaded to the fans to trust him. I think he deserves that trust.
In short, perhaps one can pose the question: If Vieira had remained at Highbury, could Arsenal change the way their football is played? I think when Wenger realized that the answer to that question was NO, Vieira was on his way out.