I wouldn't say it came too fast. Several years in a row Spurs ended up above Liverpool, but it is not like Liverpool imploded because of their success. Since 05/06 Spurs have been in the top 4 discussion with a couple of hick-ups. Pochettino took it one step further - I absolutely agree to that, but it is not like there wasn't a solid foundation there. You need a bit of luck as well. Ferguson talked about that many times. Solid teams are solid teams, but to win the CL for example you also need a bit of special talent. Finding that special talent is difficult. For Spurs that was Modric and Bale. For Pochettino he got a lot out of Kane and Alli for some time. Not saying Pochettino shouldn't get some credit for that, but even Ferguson couldn't make any team the best - he also needed rare talents like Ronaldo and many others to go from very good to winning the big titles. Had Spurs bought Salah instead of Liverpool do anyone believe Klopp would have been as successful? That is not criticism of Klopp, but you need a bit of luck to go along with being a good manager.
I didn't look anything up, so yeah, order of managers might be a bit wrong, but Harry leaving was correct and the apparent sinking back into mediocrity was looking likely until Poch came along.
Klopp... Would he be as successful without Salah etc.? I dunno, but the thing is he and his team are really good at identifying the players they want and need (VVD aside, everyone wanted him), but he went for Mane, Salah, Fabinho (who btw looked like a bust for half a season before settling in and playing world class football), Robertson, etc.
His recruitment is absolutely a huge part of what makes Klopp Klopp, and while I guess he was lucky in how some of his buys turned out, he didn't buy 4 fullbacks like Pep and play a numbers game, he goes for that one target and usually it works out great.
Very similar to Poch, Klopp came in and Liverpool was looking like they were heading for mediocrity (lost Sterling and Suarez, rest were having a bad season - I loved it, we beat that 3 times, including a 3-0 win away at Anfield), and it really did look bleak for them.
4 years later they are not only the European Champions, but looking in prime position to win the EPL and very much in contention to defend their CL title.
IMO what Poch and Klopp have done made these clubs jump a tier, from playing second (or third) fiddle to the best clubs, to being absolutely in that highest tier of clubs out there.