To paraphrase for this specific situation. When the Doctor says to Alex, "Don't you think she (Harriet) looks tired." It's planting the seed in Alex to question if Harriet is fit to continue being Prime Minister, or if she is 'too tired'. These 6 words spoken to Alex, Harriet's Sr. aide, inevitably leads to Harriet's downfall
I always interpreted this as Alex never telling Harriet, but as she knows something was said, it gnaws away at her confidence until she voluntarily steps down.
I think it's both. Alex has the idea that Harriet is too tired to continue in her post, and Harriet becomes increasingly paranoid because she is constantly wondering what the Doctor said, and how it was going to end her political career. Since she's paranoid, she starts acting more and more erratically, which feeds into the 'too tired' narrative, and eventually destroys her career.
"Don't you think Trump looks tired?"
https://scifi.stackexchange.com/questions/193743/how-did-the-doctor-take-harriet-jones-down-in-the-christmas-invasion
I always interpreted this as Alex never telling Harriet, but as she knows something was said, it gnaws away at her confidence until she voluntarily steps down.
I think it's both. Alex has the idea that Harriet is too tired to continue in her post, and Harriet becomes increasingly paranoid because she is constantly wondering what the Doctor said, and how it was going to end her political career. Since she's paranoid, she starts acting more and more erratically, which feeds into the 'too tired' narrative, and eventually destroys her career.