"What are you working on now? Why does that part take so long? Can we skip it altogether?" They're not stupid, they know a little bit about the path from point A to point B.
I have never given an estimate that was not later called into question and asked to be reduced. Usually much sooner in fact.
In many cases, asking for an estimate is fishing for a way to make the problem easier to solve. But they don't really want to make the problem easier to solve, they just want to get it done quicker (with no regard for whether or not it makes the next problem harder to get this problem done the quicker way.)
Really nobody is thinking this hard about it though, it's basically the "quarterly earnings report" problem. If we don't look good this quarter, then there's not much hope for next quarter, since we already told the boss this one would be done this quarter, and have started talking about what they'll want next quarter, though we're not ready to talk with you about that because they could change their mind at any time, so "93 days" sounds absolutely perfect, as you'll be ready for more soul crushing just in time for next quarter.
"But they don't really want to make the problem easier to solve, they just want to get it done quicker (with no regard for whether or not it makes the next problem harder to get this problem done the quicker way.)"
It is not just the quarter result, the managers also are evaluated based on what they did the last few months. If they miss their projects for the quarter there is no promotion, no pay raise.
Quarterly, monthly, whatever, ...it is the same problem. I'm talking about the dichotomy between optimizing for short-term results and maintaining a sustainable long-term vision (which does not necessarily devolve into a failed software project / team diaspora.)
If you have a good manager, you don't want them to be promoted because then it's a crap shoot whether you get another good one or not :) of course the pay raise is a compelling argument
I was only arguing against the idea that you can successfully keep your manager in the dark, you don't want that. You want a good manager who understands the problem domain well enough to anticipate these negative outcomes and who won't fixate on quarterly (or short-term) outcomes at the expense of long-term viability.
(A good manager also optimizes for your long-term success as well as their own!)
And it's not to say that optimizing for short-term goals is necessarily wrong every time. There's YAGNI – which is a perfectly reasonable argument to make at any time, but sometimes You Are Gonna Need It and there's an objective case in favor of not taking the shortcut because it will cost you in easily predictable ways.
In your scenario, the "better" way appears to require the same time anyway so just present that as the only way to do it.