Exactly. No manager is going to let their team take the foot off the gas until his team is mathematically safe, and even then each position is worth plenty of dough so they keep fighting. Beating Burnley home and away is survival and a push towards the top half.
If there was the slightest hint of Gerrard wanting Villa to go easy on either Burnley or the reds in the run in he’d lose his job and his career would be in tatters. No club or dressing room is going to tolerate a manager with interests elsewhere.
With Klopp’s new contract even if he viewed Villa as a stopgap they are the only gig in town for him now. He isn’t getting a bigger job if he messes this one up.