The answer kind of depends on how old you are. At a very introductory level, say, maybe middle school or younger, it's "okay" to refer to Jupiter as a failed star to get the idea across that a gas giant planet is sort of similar to a star in composition. But around middle school and above (where "middle school" refers to around 6-8 grade, or age ~12-14), I think you can get into enough detail in science class where this is fairly inaccurate.
If you ignore that the solar system is dominated by the Sun and just focus on mass, Jupiter is roughly 80x lighter than the lightest star that undergoes fusion. So it would need to have accumulated 80 times what it already has in order to be a "real star." No Solar System formation model indicates this was remotely possible, which is why I personally don't like to think of it as a "failed star."
Below 80 MJ (where MJ is short for "Jupiter masses"), objects are considered to be brown dwarf stars -- the "real" "failed stars." Brown dwarfs do not have enough mass to fuse hydrogen into helium and produce energy that way, but they do still produce their own heat and glow in the infrared because of that. Their heat is generated by gravitational contraction.
And Jupiter also produces heat through both gravitational contraction and differentiation (heavy elements sinking, light elements rising).
Astronomers are not very good at drawing boundaries these days, mostly because when these terms were created, we didn't know of a continuum of objects. There were gas giant planets, like Jupiter and Saturn, and there were brown dwarf stars, and there were full-fledged stars. The line between brown dwarf and gas giant - to my knowledge - has not been drawn. Personally, and I think I remember reading somewhere, the general consensus is that around 10-20 MJ is the boundary between a gas giant planet and brown dwarf, but I think it's fairly arbitrary, much like what's a planet vs. minor planet, Kuiper belt object (KBO) or asteroid.
So during Solar System formation, was there a chance Jupiter could have been a star and it failed ("failed star!") because the mean Sun gobbled up all the mass? Not really, at least not in our solar system. But for getting the very basic concept across of going from a gas giant planet to a star, calling Jupiter a "failed star" can be a useful analogy.
I don't know if it will help you, but what you are missing is the basic insight of calculus if you want. This lack of understanding generated paradoxes since the time of the Greeks. See "Achilles and the tortoise" on Wikipedia.
The basic point is that you can sum an infinite number of "intervals" (real numbers) and obtain a finite result.
For example if you sum $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$ you get 1 (perfectly finite).
The idea is the same with the deceleration. Deceleration reduces the velocity a little bit in a certain time interval, than you are farther away, deceleration becomes smaller and reduces the velocity again but a little less then before, etc. The point is the the sum of all the small reductions of velocity is finite and if this sum is smaller than the initial velocity the velocity will never reach zero and the teapot will always be flying farther away never coming back.
For example if the initial velocity is 2 and the deceleration reduces the velocity in little steps like this $2 -\frac{1}{2} -\frac{1}{4} -\frac{1}{8} -\frac{1}{16}\cdots$ the final velocity is $2-1=1$, still positive! If it started with velocity less than $1$, the velocity would become negative and the teapot will fall back on the planet. I hope it helps your intuition, but study calculus, it is useful ;)
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This is true, but you've forgotten about the Sun. Every interaction between a planetismal and Jupiter is a three-body interaction.
Above, a simulation of a low-mass planetismal moving in the effective potential in the rotating frame for a planet with mass $10^{-3}$ of its star's mass. The Lagrange points are marked with $\color{orange}{\times}$. The particle starts at $%(0.83,0.47)$ some random place I clicked; it moves ahead of the planet for two or three orbits, pausing at a couple of unstable stationary points in the rotating frame, then has a close interaction with the planet. In this case the close interaction doesn't lead to a capture, but you can see from the inset that the interaction is chaotic: it's extremely sensitive to the details of the closest approach. You can surely imagine a three-body interaction that ended in the particle being captured by the planet, even if I haven't hunted for one to show you.