Here's the reality, and maybe you've forgotten: The engineers are the ones who are going to get dumped on, and you and the VP's are the ones who are going to reap the benefits. Never forget that.
Not that I'm claiming that all executive decisions are clueless, most people are just trying to do a good job with limited information, and mistakes happen. But you run into enough executives without respect that runs in the other direction, and you start to question all deadlines.
(Which isn't necessarily a bad thing. In a healthy environment, questioning the feasibility of a deadline can lead to appropriate descoping of overreaching features, reduction of risk, etc. But the situations that lead to this kind of suspicion are rarely healthy to begin with.)
This is everything that is wrong with software.
What is software really? It is translation of business process and need into code - I spent the first 10 years of my work life mastering code, I will spend the rest of it understanding whatever business I'm working for.
Assuming you have some experienced engineers (not mangers) on your staff then having the vp set a date (and defacto choose the project) is absolutely backwards.
Bring the problem to your engineers (money) and don't look for a moon shot look for 10 smaller projects that get you 1/2 way to the goal in 1/10th the time. Don't think of it as a "hack day" think of it as a hack week or hack month.
If you have a large project and it is to "save the company" (or push it forward) and your treating it like a must do rather than a foundation (and taking care) your creating debt and starting a cycle that you will never pay off.
This is the same type of argument that makes short-term profit take precedence over long-term success. Why in the world are we getting a deadline of 4 months? You, as an executive, had to know burn rate was blowing through cash a lot earlier than that.
Engineers hate releasing poor quality products because we know we'll eventually have to fix them and we will likely be given almost no leeway time-wise to fix the actual debt, as time will always favor ugly hacks that "work" over well-written code that prevents the problem in the future.
The tech titans aren't monoliths with a collective hive-mind. Each business unit and team is responsible for projecting revenue targets and then hitting them, same as any other kind of business. And, yes, if the business unit managers fail to hit their financial goals too many times, their project will be cut and the team reassigned or laid off, same as a startup running out of financial runway.
Engineers hate releasing poor quality products, but the reality is that customers often will buy and use something laden with defects rather than nothing at all. And in any case, after the initial botched release, you can always fix stuff. See Apple.