Just because a series is endless, it does not mean it is 'infinite'.

Erm... yes it does. That is the meaning of infinite. I'll attempt to understand your distinction for the benefit of the discussion though. I imagine that you consider these objects to be finite sums that don't have an endpoint. This explains your fixation with partial sums, as to you it appears that these sums are partial sums. I'm not sure where we can go with that, because it sounds bonkers to me.

... To me, this appears to prove that the object cannot exist in 'its entirety'.

Well a line is infinite (or at least, it is endless), but we have no trouble representing the entire line as a set $\{\vec a + t \vec b\}$ or an equation $(y-y_0)=m(x-x_0)$, and we quite happily consider limits to be an object despite the fact that their construction explicitly ensures that they have no endpoint. Infinite "sums" need be no different, especially if you can stop thinking of them as sequences of addition operations and instead think of them as objects.

Why not avoid these problems and simply treat an endless object as an endless object (like I do), rather than assume 'infinity' must exist?

Actually, I am not assuming that "infinity" exists (other than as a token that is assigned to infinite sums that cannot be assigned a finite value). I am simply considering them to be an object. I call them infinite (or endless) objects as a way to differentiate them from the finite sums that are subject to different rules such as associativity and commutativity.

I keep stressing over and over that an endless series (where the endless terms are non-zero) cannot have a sum. If anything, I am fixed on NOT evaluating the sum

And yet you persist in evaluating partial sums and claiming that they have a direct relation to the infinite object.

these methods claim to 'remove the infinite parts'

I don't agree with this. I think that it's Numberphile's way of simplifying the idea for a more general audience. Either that, or the physicists interviewed don't understand the mathematical basis of the ideas. The idea of "removing the infinite parts" suggests that you have been adding the elements of the sum. My main reference on this is

Carl Bender's lectures, where he does say that adding up the terms of infinite series is stupid.

If you consider the graph of the partial sum expression, then for even values you will get a reflection about the y-axis but for odd powers you will not.

I'm actually quite interested in your graph - although not for the reasons you are. Like you, I suspect that it might give some insight into why the values are as they are, but I am curious as to whether that might lead to some realistic understanding of the physics in which this stuff is used.

"We are not content with just saying that there is some magic over there, there is magic, but we always want to explain it"... "does it mean that in some sense it is, there is a context in which this sum, this infinite sum, is mysteriously minus one twelfth? I'm not sure."

The first quote sounds like it is a reference to

Clark's third law. The second is essentially what we are talking about. There are obviously very good reasons to suppose that the sum of a bunch of positive integers can't be a negative non-integer, but the Reimann sphere suggests possibilities for why it might. My position at the moment is that we are not adding an infinite number of things together, so any rules of closure don't apply. But then, we already know that the limit of a sequence of rationals can be irrational, so why should such rules apply?

Because if you then proceed to use it, you will not understand what you are doing. And if you don't proceed to use it, then it is of no value.

Well, physicists openly admit that they don't know why it works, but they also point out that the predictions made using this stuff are good. They hope to understand it later. I'd agree with that - this stuff is close to frontier of mathematical understanding to, so there's every reason to expect that it should be incompletely understood. But I'd also say that, in a purely mathematical context it really doesn't matter. Abstract Algebra is utterly devoid of meaning to me, but that doesn't stop me from using numbers, nor does it make mathematics devoid of value. The undefined terms of any system (such as the lines and points in geometry) explicitly do not have any intrinsic meaning. We impose a meaning when we identify a relationship between them and something that we wish to study. And this relationship then means that all theorems involving those objects ought to have a meaning as a result, but it is far from obvious in all cases what that meaning might be.

But what method have you used to get this 'value' of 'infinity'?

Well the derivations goes:

We are going to assign a value on the Riemann Sphere $s$ to $1+1+1+\ldots$ using generic summation. Thus $$\begin{aligned}s &= S(1+1+1+...) \\ s &= 1+ S(1+1+1+...) &\text{by the first rule I gave above} \\ s &= 1 + s \\ \implies s = \infty\end{aligned}$$

Now, there's a lot of "fudge" in that. But the point is not in how we get to that answer, but the fact that we accept the answer that we get. Essentially it means that "this technique is not powerful enough to assign a finite value to this series". It turns out that analytic continuation

__is__ powerful enough to do that, and gives us the value $-{1 \over 2}$.

In essence, I believe that you need to stop concentrating on finite partial sums. They work well as a tool for convergent series, but are not powerful enough to assign a finite value to divergent series. I also believe that you need to accept mathematics for what it is, rather than for how it can be interpreted.