I think #1 should be read "whatever work ends up being done takes the time it takes, not influenced by the estimate" which is true. What you're saying is "the estimate influences what work ends up being done" which is also true.
In other words, there's no conflict there, just conditioning on different things.
I think GP's point is a bit different: that different estimates will result in different type and different amount of work being done for the same deliverable. This is well-known when you've overestimated - c.f. the Parkinson's law GP mentions - but they also have a good point about underestimates, which is something I haven't thought of this way.
Consider an imaginary deliverable that should objectively (from God's point of view) take about 2 months to deliver, and different estimates being made and committed to in alternate realities:
- 6 months - delivered near the end of those 6 months, possibly to above average quality, with surplus time organically divided between improvements elsewhere in the project, other tasks, research, slacking off, and general waste.
- 2 months - delivered exactly on time, average quality, team fully focused on task.
- 1.5 month - delivered exactly on time, average quality, team fully focused on task, if a little tired.
- 1 month - delivered exactly on time, below-average quality, the team tired, demoralized by the anxiety and stress of meeting an unreasonable deadline.
There's indeed little difference in work being done if the estimate is in range of ~1.5 months to 3 months. Above, Parkinson's law will noticeably kick in. Below, corners will be cut, people will burn out, and this unsustainable approach will start as soon as people realize the deadline is unrealistic - which may be at the very beginning.
In other words, there's no conflict there, just conditioning on different things.