The way you are talking about jargon in mathematics suggests you have a limited experience of what mathematicians do. Here's an illustrative example of mathematics as done by mathematicians.
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Define: An integer n is `even' if there exists some integer m such that n = 2m.
Theorem: For any two even integers n and a, the sum n + a is an even integer.
Proof: Since n and a are even there exist integers m and b such that n = 2m and a = 2b. Now,
n + a = 2m + 2b; by assumption
= 2(m+b); by the distributive property
= 2z; for the integer z = m+b
Therefore there exists some integer z such that n+a = 2z. Hence n+a is even.
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That is to say, in mathematics we introduce some definitions/gibberish/jargon (in this case `even') and then we use logic to reason about the implication of our choice of definition (the sum of even integers being even.)
The important thing is that the definition plays an essential role; definitions are the building blocks on which all of mathematics operates. To emphasize: if you strip away the definitions you literally have nothing to build on - we can't apply logic to nothing and arrive at something.
This leads to the point I made in my earlier comment: the reason we need definitions rather than intuitive explanations is that you can't logically reason about a concept unless you nail down the relevant details of what that concept is exactly. We can't do the 'proof the theorem' part of the above example.
So how does mathematics then fit into application?
Guy 1: In this basket I have as many stones as I have fingers and in that basket I have as many stones as I have toes. For each basket I can pair up the stones so that each has a partner. Will this still be the case if I combine the stones from each basket?
Mathematician: Well, lets represent the number of stones in each basket with the integer 10. Pairing stones corresponds to the integer being even and combining the baskets corresponds to adding the two integers. I note that 10 is even since 10 = 2x5 and so I can apply my theorem to conclude that the sum 10+10 is even. Thus I conclude that when you combine the baskets you will still be able to pair each stone with a partner.
Guy 1: Wait, wait, wait! I don't understand this 'even' jargon. Do it again without the jargon.
Mathematics: The definition of 'even' was central to my whole processes. Without it I can't even set up the problem, let alone apply the theorem used to justify the answer. I could perhaps just give you an answer, "MATHEMATICS SAYS YES", but then you wouldn't be able to repeat it yourself for different numbers of stones.
If the above is understood then I can quickly address the claims you have made.
> it's better to generalize upwards from reality and actual use cases instead of starting utterly removed from reality and trying to apply the generalizations downwards.
Mathematics is generalisation utterly removed from reality. This is why we have "Adding integers" and not "Adding together collections of dogs" and "Adding together collections of apples" and "Adding together collections of hats" and ...
> Sure, it's convenient for a mathematicians to be able to use shorthand gibberish to talk to other mathematicians.
Mathematics is the practice of defining new gibberish and then reasoning about that gibberish. The gibberish isn't a shorthand for something, it is the thing.
> It doesn't justify pushing this jargon on other fields.
Mathematics is definitions/gibberish/jargon. Applying mathematics to a field thus means applying definitions/gibberish/jargon to that field.
> When it comes to a point where gibberish becomes the only way to explain mathematical abstractions, then you should step back and ask yourself "where the hell did this go wrong?".
At least since Euclid's formulation of geometry.
>> So by the intuitive explanation we can make this single line any dimension that we want.
> And if this makes sense in the given context - sure, why not?
The problem is that it doesn't. Your explanation of an n-dimensional space is more a description of the larger space in which our space of interest is embedded.
In all instances the space (the line) remains unchanged, the only thing which changes is how we are describing it. For the dimension of the space to be a property of the space it needs ignore how we choose to describe it.