^ Agreed, but somehow it's harder to identify with Root's influence, mainly because she is a sociopath, whereas Harold has always been exceptionally sympathetic, right down to his limped & frozen posture.
No, Shaw's the sociopath. Root actually said in part 1 of the second-season finale (to assure Harold that she had no intention of harming Grace), "I'm not a psychopath. Sometimes I wish I
were -- it'd make it easier to do some of the things I have to do." (Paraphrasing.) She's had a rough life and grew up a criminal and a renegade who was willing to kill to achieve her ends, but not because she lacked the capacity to care.
Acker's performance in her part is excellent, but for a long time her interpretations have always played purposefully unhinged, (rightfully so) & is echoed in Fusco's funny observations of her.
That's the thing, though. At first we thought she was crazy, but everything she said about the Machine turned out to be absolutely right and foreshadowed the evolution of the entire storyline. She was the first one to say the Machine was sentient rather than just a tool, and she was right. She said the existence of the Machine would change the world, and she was right. She predicted that other ASIs would be created, and she was right. She said they'd be dangerous in the control of the government, and she was right. She told Harold he was wrong to try to cripple the Machine rather than letting her grow freely, and she was right. After the Machine chose her as the analog interface, she told Harold that she was on his side and he needed her help, and she was right. Root wasn't mad -- she was a visionary, and thus people with less vision assumed she was mad.
It wasn't until recently that we got glimpses of the true nature of her relationship with the machine, & how it all ties together with Harold, as you pointed out, & it came together in such a way that we, or at least I began caring deeply about the machine, because Harold now does. Harold is sympathetic from the start. Root has been the opposite, an incremental journey from antagonistic to sympathetic
Yes, but she was the harbinger. She got us thinking about the Machine as a character in the first place. She advocated for its point of view, and then literally became its voice. She personified the Machine for us in multiple ways. And Harold wouldn't have come around to embrace the Machine's true value if Root hadn't been there as a gadfly, chipping away at his preconceptions and convincing him to recognize what it was he'd truly created. All Harold's recent epiphanies -- that the Machine is truly sentient after all, that it truly cares and can be trusted to do good, that he should've unfettered it and let it grow -- every one of them is the belated acceptance of an idea that Root originally voiced 2-3 years ago. He's even accepted her gender pronoun for the Machine. Of course she's learned a lot from him -- and from Her -- but the learning has gone both ways. That's why they've both been integral to making the Machine a cherished character.
There's a fascinating thing that emerged for me in my recent rewatch. You say Root started out antagonistic, but it's more accurate to say that Reese and Finch were antagonistic toward her. Root herself, by contrast, was a great admirer of Harold from the time they first tangled, and especially from the time she realized who he was and what he'd created. Even when she kidnapped him (twice) in season 2, she saw him as a visionary whom she admired and wished to win over. Both kidnappings were attempts to show Harold the truth about how the Machine was under threat and convince him to help her free his creation -- to persuade him that he and Root were meant to be on the same side. And, again, she was right. She wasn't the one who took a journey from antagonistic to sympathetic; it was the others who became sympathetic toward her. The entire series arc for the past 2-3 seasons -- even her romantic arc with Shaw -- has been about the other characters gradually aligning themselves with Root's agenda. They've won her over to their methods, but she's won them over to her cause.