Rodgers did very well to capitalise on the perfect storm that was the 2013/14 season. United utterly collapsed post-Fergie, Chelsea were yet again starting a season with a new(ish) manager who repeatedly said they weren't going to win the league, and Arsenal and Spurs were Arsenal and Spurs, with a resurgent City under their new manager being the only one to top them. Liverpool had no European distractions, and Rodgers utilised a system that was very effective for 95% of the season, that was built around Luis Suarez who was starting to emerge as a truly world class player.
After falling just short of winning their first league title in forever, expectations had (perhaps unfairly) been raised to a level that they'd not been at for a number of years. Their last title challenge had come out of the blue in 2008/09, and that was also the last time they'd qualified for the Champions League. It was obvious that Rodgers had failed to adapt Liverpool's tactics post-Suarez to be as effective as 2013/14, but that would happen in any team so reliant on the ability of one player when he left. The board gave Rodgers a summer transfer window to add strength to the squad after missing out in 2014/15, only to sack him 8 games into the 2015/16 season. If the board wanted a reason to get rid of him, they weren't going to get much better than missing out on CL football, and only winning 2 of their last 9 games, including a 3-1 home loss to Palace and a 6-1 away loss to Stoke in their final two games.
My personal feeling is that the Steven Gerrard Final Farewell Tour overshadowed the ambitions to retain their CL spot in 2014/15, and was a huge contributing factor in their end of season slump. When efforts should have been focused on getting as many points as possible during the run in and potentially winning their first bit of silverware since Dalglish's League Cup, all the focus was on Gerrard leaving and the possible fairytale ending of him winning the FA Cup one more time whilst also leading his team to CL qualification. This forced Rodgers into something of a no win situation. He either played an aging and increasingly useless Gerrard to go with the fairytale, or didn't play him and ruined the farewell of a club legend.
Gerrard started 16 of Liverpool's first 19 games, and just 4 of the next 10. In the 20 games that Gerrard had started at this stage, Liverpool won just 8, drew 5, and lost 7. In the 9 that he didn't start, they won 8, drew 1, and hadn't lost any. With 9 games left to go, Liverpool were 2 points off 4th, had been unbeaten in 13 (taking 33 from 39 points), and were on a 5 match winning run. Their 30th game was the game against United at Anfield, which is when the SGFFT began. As we all fondly remember, he came on as a half-time substitute to turn the game in Liverpool's favour, then got himself sent off less then a minute after the restart.
Liverpool won just 2 of their final 9 games and ended up in 6th, 8 points off 4th. By the end of the season, Liverpool's league record for games that Gerrard started was P25 W9 D7 L9, and P13 W9 D1 L3 for games that he didn't start. For what it's worth, the 6 points dropped during the unbeaten run were dropped in games that Gerrard started. Percentage-wise, Liverpool won just 36% of the games he started, and 69% of those he didn't.
As mentioned, focus had shifted from Liverpool qualifying for the CL and winning the FA Cup, as they were potentially on course to do, to Gerrard qualifying for the CL and lifting the FA Cup in one last heroic effort on his part. His inclusion in the team was detrimental to their league form, an the same happened in the FA Cup. Gerrard started just 3 of Liverpool's 7 FA Cup games that season. The first was the 3rd round tie against League Two AFC Wimbledon, which they scraped 2-1, and the second was 4th round replay against Championship Bolton, which they were actually losing until the 86th minute, not picking up the winner until the 91st, despite playing the last half hour against 10 men. The third and final FA Cup game that Gerrard started was the semi-final against Villa (which they lost), which happened to mark Gerrard's return from suspension following his sending off against United.
I think Rodgers can rightfully feel hard done by. He was given 46 games (split 38/8 between seasons) to prove that he could reestablish Liverpool as CL regulars and serious title contenders following the surprise of 2013/14, and had to cope with their best player in decades and the focal point of their team leaving, and a club legend hijacking the final part of the season to wish himself a fond farewell. He was sacked very early on in 2015/16 as it looked like the season's targets were already slipping away (seemingly with little consideration of how close they actually were to qualifying the previous season before Stevie Me stole the spotlight). Klopp was brought in to put things back on track, and is now 54 games in (split 30/24 between season), with Liverpool having finished 8th in his first 4/5 of a season (and Klopp's points per game after 30 games being just 0.1 better than Rodgers' was after 8), and this season, despite having no European football, United, Chelsea and City having new managers, and Arsenal and Spurs still being Arsenal and Spurs, they find themselves out of every cup and 5th in the league, with 0 wins from their last 5 league games, with that standing at 1 from 10 in all competitions (with the 1 being an FA Cup replay against League One Plymouth that now counts for nothing because they got knocked out by Championship Wolves).
I think anything Rodgers does domestically with Celtic should be taken with the largest pinch of salt because winning everything is the expectation with Rangers being shite, and I'm not convinced that he's actually good enough to reestablish Liverpool as a CL regular and serious title contender. That said, I'm not sure anyone is. The expectations following 2013/14 were and are silly, particularly in the time frame he was given and with the transfer budget Liverpool afford their managers. However, the goalposts appear to be shifting again following Liverpool's recent dive in form, and simply challenging for top 4 is seen as sufficient. Rodgers was sacked despite doing just that, and given almost no time to prove he could do it again.
I don't think he was as special as Liverpool fans made out during 2013/14, nor do I think he's as bad as people made out (myself included) at the time of his sacking. He's more than good enough to keep Celtic on top in Scotland, and a decent manager for mid-table PL side. I'm not sure he's got the strength of character or tactical nouse to be any more than that. Likewise, I don't think Klopp's as special as people made out during Dortmund's successful run, nor as poor as he looked during the first half of his final season at Dortmund, and has looked over the past month. I think he's an incredibly one-dimensional manager in terms of the tactics he employs, and his public persona/personality and the way he handles himself with the media is prone to backfire. I think both have enjoyed purple patches aided by a player, or players, emerging as world class talents under their management, with Klopp's purple patch obviously lasting longer and bringing in more success than Rodgers. How much they had to do with the emergence and development of such players remains to be seen as they manage at different clubs.