I think the most helpful thing to tell you here is that energy is not a stuff, not in the ordinary sense. Let me give you two definitions of stuff-ness:
Draw any imaginary box in space. Everyone agrees on the quantity of stuff inside that box, and it doesn't change arbitrarily, but mostly just by stuff going through the boundary of the imaginary box.
Draw an imaginary box im space, different people disagree on the amount of stuff in the box but everyone agrees on changes and how much stuff flows through the boundary.
All 1-stuffs are 2-stuffs automatically, it's a looser definition. Think of an example, toys in your niece's toy bin are a 1-stuff, everyone agrees in principle on the count of toys in that bin, we may have uncertainties about it or might happen to be in the wrong position to see into the toy bin. But the number of toys is some objective $N$ that hopefully we could all agree on.
Contrast this with the amount of forward momentum inside a train compartment. This is a 2-stuff. Everybody agrees that to change it you need to change a velocity, and all reference frames agree on changes in velocity. However the absolute amount of forward momentum appears to be near zero to someone who is inside the train and sees everybody sitting at rest, but appears to be very large to someone who is watching the train pass by from the ground.
There is no harm in pretending that a 2-stuff is a stuff as long as we remember that not every reference frame agrees on the total amount of stuff in the box, it is more than, as you say, “a numerical value with no real meaning” as it tracks with the stuff going through the boundary. So in the momentum case, these changes in momentum flowing out of the train compartment, correspond to a Newton's third law Force-pair where one of the objects is inside the train and the other one is outside. And what could be more "real meaning" than a force like that? An irregularity in the tracks knocks you and everybody else off their feet: surely that has real meaning!
Well, energy is precisely this sort of 2-stuff. Consider $N$ masses $m_i$ with velocities $v_i$ in the center of mass frame $\sum_i m_i v_i = 0,$ so in the center of mass frame the momentum is zero and the energy is in general nonzero, it is $$K = \frac12 \sum_i m_i v_i^2.$$ If I am traveling at velocity $u$ relative to this frame I see a kinetic energy
$$K' = \frac12 \sum_i m_i (v_i-u)^2 = K + \frac12 M u^2$$ where $M = \sum_i m_i$. In other words I see the center of mass kinetic energy plus a constant. Since no internal process of that cluster of particles will change $M$ or $u$, every energy difference that the center of mass frame sees, I also see. Furthermore energy is conserved, so it is a 2-stuff proper.
But, everyone is measuring the amount of work the system could do if it were elastically brought to absolute rest in their frame of reference, this is, as the above expression says, equal to the work to bring the system to rest in its center of mass frame plus the work to bring the center of mass frame to rest in my frame, $\frac12 M u^2.$ We all disagree because we disagree on this value.
It is something like if each toy in the toy bin had a little price tag on it and we measured the box by what was really important to us, which is the amount of value we can get by selling all of these toys, but we all had a different corporate bonus that we would get if we managed to sell the toy bin itself, which is only possible if you sell all of the toys in it. It is an imperfect analogy but maybe it helps. Those dollars are real, in this weird hypothetical they correspond to money that would be in my bank account at the end of the day, makes a very real difference to me. But the actual amount is different for different people looking at the same toy bin.
I hope that helps to see why differences in energy are "real" and energy itself is helpful even if subjective because it helps us get to the real things.
