That’s not the only reason at all. Klopp was clearly making progress from early on. They finished 6th in Rodger’s last full season, 8th the season Klopp replaced him and have been top 4 ever since. He made them top 4 in his first full season after the club missed out on CL football in 6 out of 7 seasons before that. If Jose does the same, he won’t have a problem.
He’s had the Coutinho money to reinvest which won them a CL and Premier League which then gives him more money to invest.
His style and demeanour have helped because it’s easier to buy into it rather than Jose’s attritional, defensive style. But, like I say, if Jose gets us top 4, keeps us there and wins trophies like Klopp has done at Liverpool since his first full season, Jose won’t have a problem.
A team that plays attacking football and doesn’t win trophies (eg Poch’s) is far more lovable than one who defends and wins nothing because people like being entertained. But win trophies playing defensive football and you’ll also be remembered fondly. The problem is, it’s hard to see it at the moment.