He got ‘some’ games and had a terrible record under Sherwood. Check his stats. It’s simply a lie that Poch didn’t make Kane the player he is today.
Up until April of that year, he featured mostly as a sub towards the end of games, and it's true his record as a sub wasn't all that. In his 6 starts in the league that season, however, he scored 3 and got 2 assists. That's not a terrible record.
You have picked selective games from the last season. Rodgers had them almost win a title which hadn’t been for a good 20 years. They finished second but were no where near under Rafa or Houdlier. The season after the Suarez season they were something like 8th in the league when Rodgers got sacked. Exactly the same situation Spurs have put themselves in. Check some facts before trying to look at a few results.
Rodgers finished 2nd on 84 points, 2 points off City. Rafa finished 2nd on 86 points, 4 points off United. Rafa got about as close to winning it as Rodgers did, I'd say.
Liverpool were 10th in the league when they sacked Rodgers, having finished 6th the previous season. They were one point better off than Poch's Spurs were at the same point this season, but had scored less and conceded more. They had 5 wins in their last 16 league games, and three of those came against relegation fodder, while one came against a newly promoted side. In those 16 games, they had scored 17 and conceded 27. Their form was dire, and showed no signs of improving. So your claim that Liverpool weren't doing that badly at the time is clearly false, as
@lysglimt correctly pointed out.