My recollection is that there were various reasons given for him leaving - wife didn't settle, sectarianism, family etc. I think it was more simple than that - it was down to success or otherwise!
It greatly pains me to say this, but whilst John Greig was one of the greatest Rangers players ever, his record as manager was really poor though the resources made available to him by the board were pretty modest. In his 5 years or so as manager, from 1978 to 1983, he never won the league. In fact, his record was 2nd, 5th, 3rd, 3rd and 4th. Bett was a great footballer but not the kind of player to drag that team up by its bootlaces. After Greig stepped down, Rangers league performances continued to be poor - 4th, 4th and 5th in 83/84 through to 85/86.
Bett joined Aberdeen in 1985 when Rangers were still just as poor as they had been when he played for them. I'm not surprised that he chose an Alex Ferguson led Aberdeen over Rangers of that era. Souness, like Greig, was untried as a manager, when he joined in 1986 (after Bett had gone to Aberdeen). Having already played under an inexperienced and unsuccessful manager in Greig, why would he risk doing so again? Hindsight is a wonderful thing but the Rangers team (and club) inherited by Souness was simply awful. Looking back, Bett probably knows he missed out but looking forward at the time, how would he know that Souness would reverse nearly 10 years of neglect so successfully?
In truth, as good a player as he was, Bett probably wouldn't have been a first pick in the team put together by Souness.