I think everyone was briefed quite widely it was because of the quite severe reputational damage to the brand that he had caused with how he handled the racism within the team/club. I don't think it was feasible for him to continue from a PR perspective.
You'd also have to acknowledge that Dalglish was only ever really a post Hodgson stopgap. FSG place no importance on cups and wanted to see the team in the CL, or getting closer to it, and they were going backwards under Dalglish, finishing I think about 25 points off the top 4. Rodgers was very much a manager who fitted what they were trying to do, he was a young, bright coach who had a track record for developing and improving younger players and they could see he would be a much better fit for what they were trying to do.
It's extremely misleading to make out he was sacked for coming 7th. In the same was it's a bit misleading to say that we sacked Allardyce for coming 8th, a if it's some sort of urinating in the wind contest. Neither manager were the right fit for what their respective club/management were trying to achieve, and while they both did reasonable jobs as stopgaps they were never really seen as any more than that.
So, with respect, I think your analysis of what had occurred is a little short sighted in the way it was presented.