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Yes, the fundamental reason is conservation of energy. If there is not enough energy stored in the rocket's fuel to escape the gravity well, it doesn't matter whether you expend it all at once (and fail to reach escape velocity) or expend it over time as in your example. "Escaping the gravity well" from any point takes a well-defined amount of energy (which we call "gravitational potential energy") so your fuel use strategy won't help you in the end.

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Yep. Along similar lines of thought, there's the issue that the equations and laws governing classical mechanics fall apart in such settings.

Even so, assuming that we can apply classical thought to the problem: the simplest roadblock to rockets escaping from a black hole is that their change in mass per time unit multiplied by their velocity must at least partially exceed the gravitational pull.

Either you utilize ungodly amounts of fuel per second or you already have a crazy-high velocity or some combination of both. The math just makes it infeasible. There's limits on mass you can have and eject, and limits on attainable speed.

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