While we can look at Newtonian physics and see that an object with escape velocity greater than $c$ would give rise to many of the properties of what we now know as blacks holes, the behavior of black holes is more complicated than what is visible from a Newtonian perspective. The fact that objects can't escape from a black hole is much more than just "everything is travelling slower than the escape velocity".
Even if it weren't from the general relativity effects, special relativity effects such as objects gaining mass-energy as they accelerate, and mass-energy approaching infinity as an object approaches $c$, would prevent an object from escaping. Again, this doesn't take into account the actual general relativity effects, but since you presumably aren't familiar with that level of physics, the following addresses why even without them, your idea of a rocket escaping a black hole doesn't work.
A stone with a thruster with infinite fuel can travel as slow as 1 m/s and still escape Earth.
Infinite fuel is not possible. So I'll interpret this as "sufficient fuel". How much is sufficient? Well, if it were already travelling at escape velocity, then a certain percentage of its mass-energy would be made up of its kinetic energy component. Call that percentage $p$. For a rocket starting at zero velocity to escape Earth, the percentage of its mass that is fuel must be at least (in reality, due to inefficiencies, much more than) $p$.
As an object approaches the speed of light, the percentage of its mass-energy that is kinetic energy approaches 100%. So if the escape velocity of a body is $c$, then a rocket starting at zero velocity at its surface would have to be made up of nothing but fuel and have a way of converting that fuel to kinetic energy with perfect efficiency. Once the escape velocity exceeds $c$, more than 100% of the rocket would have to fuel.
This shows that this sort of analysis breaks down when you have an object massive enough to be a black hole. To understand exactly how it breaks down, you need to understand general relatively, but at the very least this shows that what you propose doesn't work.