What is the probability for an electron of an atom on Earth to lie outside the galaxy?

The quantity you should consider first is the Bohr radius, this tells you an idea of the relevant atomic scales,

$$ a_0 = 5.29\times 10^{-11} ~{\rm m} $$

For hydrogen (the most abundant element), in its ground state, the probability of finding an electron beyond a distance $r$ from the center looks something like (for $r \gg a_0$)

$$ P(r) \approx e^{-2r/a_0} $$

Now let's plug in some numbers. The virial radius of the Milky Way is around $200 ~{\rm kpc} \approx 6\times 10^{21}~{\rm m}$, so the probability of finding an electron outside the galaxy from an atom on Earth is around

$$ P \sim e^{-10^{32}} $$

that's ... pretty low. But you don't need to go that far to show this effect, the probability that an electron of an atom in your foot is found in your hand is $\sim 10^{-10^{10}}$.

What is said in the video is true, but... remember that the atomic theory is just that: a theory. The theory itself predicts that perturbations will have a really big influence on the results.

Take into account that the models are based on hypothesis, which are easily violated. For example, spherical symmetry, which allows finding the solution in the hydrogen atom (or more accurately, the Coulomb's potential in QM). Reality is never like that, but we can say that "it is close enough" if the atom is far enough from other objects.

Nevertheless, from here to outside of the milky way there are so many perturbations that the model would just fail. You can say that there's a level $n=1324791$, but there are so many particles out there that the effect of your atom is absolutely beaten by ANY other.

So, does it really make sense to calculate such probability if anything can capture that electron much more easily? I don't think so.

The way you phrase your question violates quantum mechanics: saying "there must be a portion of all atoms on Earth whose electron lies outside the Milky Way" is not a statement that makes sense within Quantum Mechanics. What you can ask, and what others have answered, are variations of the question of how probable it is to find a bound electron at galactic distances from the nucleus it is bound to.

I'm emphasizing this point which we would usually dismiss as semantics because this distinction makes it easier to understand that there is a second way in which your question doesn't make much sense besides as an exercise in the numerics of exponential functions: electrons are indistinguishable. How do you know that the electron from which the photon of your measurement apparatus scattered is "the" electron belonging to the atom? The answer is that you can't unless you know that there are no other electrons around. So you would have to keep your atom in a trap whose vacuum is such that the mean free path length exceeds the radius of your excited atom by several orders of magnitude, which implies that the trap is equally large. Actually, you wouldn't be able to do the experiment with a trap that is only several orders of magnitude bigger than the galaxy, you would actually need one that is lots and lots of magnitudes bigger. Why? Because every other electron in the universe has a non-vanishing probability to be found inside your trap and there are lots and lots of electrons. You want the total probability of hitting a stray electron to be sufficiently small so as not to perturb your experiment. Otherwise you cannot assign the electron which scattered your measuring photon to the specific atom that you care about. After all one doesn't look for an electron in any sense like one would look for a heating cushion.

Edit: I want to add two things which might be of interest if you want to dive deeper into electrons far from the nucleus.

First, you can actually find direct measurements of the electron clouds of hydrogen, see at this stackexchange page: Is there experimental verification of the s, p, d, f orbital shapes? This shows, never mind the terrible color scheme in the article, the rapid drop of the probabilities at increasing distances.

Second, atoms where the electrons are far from the nucleus are actively researched. In these so-called Rydberg atoms the electrons are excited to energy levels just below ionization where current experimental setups can get close enough to ionization to reach atomic radii $r \sim \textrm{const.}/\Delta{}E \sim 100 \mu m$ with $\Delta E$ the ionization energy. That's still a far cry from galactical distances but these experiments show that quantum mechanics actually works a few orders of magnitude closer to the length scales you were interested in.