Ownership loved Jon Lester. As far as I'm concerned any GM who couldn't convince ownership to open up their wallets to retain the services of a power ace and one of the most dependable starters in all of major league baseball, and a beloved hometown hero who had played a role in not one but two Championships and was beloved by the fans, for a price tag that it would turn out was still under market for a man of his talents, didn't do his job very well. A GM has to have the smarts and persuasive ability to prevent ownership and upper management from backing him into a corner. Even if lowballing Lester wasn't Cherington's idea, as general manager he has to be able to stop ownership from screwing up his rotation with their own bad ideas.
It the goal was to maintain competitive teams in 2015 and onward, there was no reason for Cherington to let himself get pushed around when it came to Jon Lester's salary. The contract Lester got proved to be an absolute bargain, substantially less than other top pitchers have commanded since, and I have no doubt that if we'd been in on Lester for similar money, either before the trade or in free agency, he'd be here right now.
The inability to challenge or change ownership directives when they're destructive to the team is the sign of a GM who can't command the respect of ownership and lowballing Lester to that extent shows that either he doesn't have a finger on the pulse of the free agency market, or is unable to argue against someone in the organization who doesn't understand free agent pitchers.
And even if it was ownership's doing, or Lucchino's you have to think that DD or Theo would have been able to talk them out of it. We certainly know Theo was willing to go to great lengths to try to change ownership's mind on what he felt was a bad idea, if you remember his reaction to the Beckett trade. That's part of a GM's job, to make his call and stand his ground. Needless to say, if Cherington did value Lester's role as a dependable top of the rotation starter, he didn't defend it properly. Whether through lack of understanding of the free agent pitching market, or simple lack of balls, either way it shows the lack of a skillset a good GM needs to have.
Lester was the lynchpin of the rotation, it was always going to be easier to rebuild the rotation with him than without him, and getting him for an under-market deal in which he'd easily pay for himself should have been more than feasible. You absolutely CAN make that sales pitch to an owner if you're willing to make a game effort to wear the pants in the relationship.
If Cherington couldn't do that, if he couldn't talk back to ownership and change their minds on key areas of team construction, then he wasn't actually a GM at all, much less a good one, and we're better off without him. General manager is considered an executive position for a reason, if Cherington was treating the job as advisory only, then he was overpromoted and is now back at an appropriate level for his talents.