I actually think he leaves before he's sacked. I'll say a lot of things about Klopp, but I'm not sure hes the most money obsessed fella out there. I think he also understands, waiting to be sacked and the rep completely gets trashed is never the most sensible ploy. I just don't see a world where he openy volunteers information about him resigning and mentions "for a break" if thats not somewhere in the thinking.
As you've said a lot, he's not really a long term domination guy. A liverpool fan Rory Smith wrote a good peice in the NYT this week and essentially said that. He has plan A, eventually the players get burned out and tired from plan A so then they do plan A worse, and tumble down the league. There's no phased planning, and he needs young, fit footballers to come in, which are generally very pricey.
I don't think any of them are grasping they have probably got 18-24 months of probably treading water- trying to avoid further decline, before trying to claw back a 30+ point deficit to the top of the league, which tends to take them 4-6 years. He's not hanging around for that. This also assumes the tap stays switched on, and a Hodgson style collapse doesn't happen (which essentially set them back to being 10 years off the title).
The stars are really aligning for Gerrard, he will have won his 1st trophy (at the 9th attempt in a 2 horse race) but you can see a big push for him coming soon in the Echo/BBC old posh boy Pearce in the Athletic. In truth, it would be an awful fit currently. His experience is less than Lampards, and Lampard took Chelsea backwards with huge recruitment and a world class youth set up. They have a bog average youth set up and at best very little to spend.
His "achievement" at Rangers (essentially being out thought bey Neil Lennon on 5 out of 6 occasions) is predicated on having one of the oldest and expeirenced squads in the league, and calling in the odd favour from PL academies (specifically Liverpool) or lads like Defoe. I'm not trying to be hard on the fella, but those skills wont be whats needed at Liverpool, certainly under FSG. In a strange way, he'd have probably been better doing it the internal route, learning under Klopp, managing the reserves then the 1st team. He'd have at least got used to the structures.
I think he would be an awful fit, and I'd imagine a lot of the players who've won titles and joined after he left would grow tired of the old "Im Stevie G" chat he'd be giving every 2 minutes.
My sense is, they will go Ljinders. He probably has the best chance if Klopp vacates position before the back end of the season. Loads of their know it all guys arelike "just got to get to the summer". Reminds me of that final Rodgers season when they were on the beach. The things is, there's 3 months to go. Thats a long wait to be treading water.