Consider any local field$K$, endowed with its topological field structure. We define the function $| \cdot | : K \to \mathbb{R}_{\ge 0}$ as
$$|x| = \frac{\mu(xS)}{\mu(S)},$$
where $\mu$ is any Haar measure (which exists because a local field is additively a locally compact group), and $S$ is any measurable set of nonzero measure (see e.g. Definition 12.28 here). This definition can be shown to be independent of $\mu$ and $S$.
Now consider a vector space $V \simeq K^n$ over $K$, and equip it with the $p$-length function$$\|(x_1,x_2,\ldots,x_n)\|_p = (|x_1|^p+|x_2|^p+\ldots+|x_n|^p)^{1/p}$$ for some $p \in [1,\infty]$ (where as usual we define $\| \cdot \|_\infty$ as the limit of $\| \cdot \|_p$ as $p\to \infty$). There is a unique critical value $p=p^*(K)$ for which this length function becomes especially symmetric, in that its associated group of isometries (i.e., the subgroup of $GL(V)$ that preserves $\| \cdot \|_{p^*(K)}$) acts transitively on the corresponding unit sphere. The amazing fact is that this critical value $p^*(K)$ always coincides with the degree $[\bar{K}:K]$ of the algebraic closure of $K$!
Here is a proof sketch:
From the classification of local fields, it is known${}^{(\dagger)}$ that $K$ is either $\mathbb{R}$, $\mathbb{C}$ or a non-Archimedean local field (that is, a finite extension of either $\mathbb{Q}_P$ or $\mathbb{F}_P((t))$ for some prime $P$).
If $K=\mathbb{R}$, $|\cdot|$ is the usual absolute value, and the critical length function turns out to be $\| \cdot \|_2$ (the usual Euclidean length), invariant under $O(n)$. On the other hand, $[\bar{\mathbb{R}}:\mathbb{R}]=[\mathbb{C}:\mathbb{R}]=2$.
If $K=\mathbb{C}$, we have $|x|=\bar{x} x$, and the corresponding critical length function is $\| \cdot \|_1$, invariant under $U(n)$.${}^{(\ddagger)}$ On the other hand, $[\bar{\mathbb{C}}:\mathbb{C}]=[\mathbb{C}:\mathbb{C}]=1$. Note that neither $|\cdot|$ nor $\|\cdot\|$ satisfy the triangle inequality in this particular case (this is why I avoided using the terms absolute value and norm throughout the question).
If $K$ is non-Archimedean, $|\cdot|$ is a suitably normalized $\chi$-adic absolute value for some $\chi$, and the corresponding critical length function is the supremum norm$\| \cdot \|_\infty$, which can be shown to be invariant under $GL(\mathcal{O}_K^n)$, where $\mathcal{O}_K$ is the ring of integers of $K$. On the other hand, it is known that the algebraic closure of $K$ has infinite degree, i.e. $[\bar{K}:K]=\infty$.
Thus in all cases we have $p^*(K) = [\bar{K}:K]$.
A problem with this proof is that it gives no indication as to why the relationship holds: after all, the values could just happen to be the same by complete coincidence. For that reason, I am wondering if there exists an alternative proof of this fact that is "$K$-agnostic", i.e. a proof that does not use at any point the Artin-Schreier theorem, nor the classification of local fields, nor any properties from a specific such field (such as the existence of a symmetric/Hermitian inner product) other than the degree of an algebraic closure and properties derived from it.
Such a proof would allow one to meaningfully speak of a relationship between those two quantities, and to state things like "in a field with a $3$-dimensional algebraic closure, the natural analogue of the Pythagorean theorem would be $|a|^3+|b|^3=|c|^3$" without appealing to the principle of explosion. I know it is hard in general to formally determine whether a proof does or does not use a fact, hence why I'm labeling this with the soft-question tag, but hopefully the kind of proof I want is clear, at least informally.
In summary, my question is:
Given a local field $K$, with $| \cdot |$ defined as above, is there any $K$-agnostic proof that the group of isometries of $\|\cdot\|_p$ (defined in terms of $| \cdot |$ as above) acts transitively on unit spheres if and only if $p = [\bar{K}:K]$?
It seems reasonable to try proving first that $p^*(K) = 1$ iff $K$ is algebraically closed. That is where I am currently stuck. The thing that makes linear algebra over an algebraically closed field "special" is that every linear transformation has an eigenvalue, but I am not sure how to apply that fact to show that the critical length function is $\|\cdot\|_1$.
UPDATE: As suggested by Torsten Schoeneberg in the comments, perhaps it will be easier to prove the relationship $p^*(K) = [L:K]\: p^*(L)$ for a finite field extension $L/K$. Consider such an extension with $[L:K]=n$, and suppose we know $L$ has a critical $p$-length function for some $p$. Now, $L \simeq K^n$ is additively a vector space over $K$, which is equipped with the product topology. This means that, if we identify $k\in K $ with its image under the inclusion $K \subseteq L$, we have
$$|k|_L = \frac{\mu_L(k S^n)}{\mu_L(S^n)} = \left(\frac{\mu_K(k S)}{\mu_K(S)}\right)^n = |k|_K^n.$$
On the other hand, note that the subgroup of $GL_1(L) = L^\times$ preserving $|\cdot|_L$, i.e. the group of invertible elements of norm $1$, acts transitively on the associated unit sphere (which coincides with the group itself) in a tautological way, since any element $x$ of norm $1$ can be sent to any other element $y$ of norm $1$ through multiplication by $x^{-1}y$. Since $GL_1(L) \subseteq GL_n(K)$, this proves that the length function on $K^n$ defined through the identification $L \simeq K^n$ as $\|\cdot\| = |\cdot|_L^{1/n}$ is a critical length function, that moreover satisfies the homogeneity property $\|kv\|=|k|_L \|v\|$ (which follows by the above relationship between the Haar measures and multiplicativity of $|\cdot|$). In a similar way, from the inclusion $GL_m(L) \subseteq GL_{mn}(K)$ we can prove that any critical length function $||\cdot||$ on $L^m$ induces a corresponding critical length function on $K^{mn}$.
The only ingredient left would be to prove is that there always exists some basis $\{1=a_0, a_1, \ldots, a_{n-1}\}$ of $L$ such that the above defined length functions are of the required form, that is, $|\sum_{i=0}^{n-1} k_i a_i|_L = \sum_{i=0}^{n-1} |k_i|_K^n$. Once we have that, we can show using isotropy subgroups that the length function induced from $L$ is critical in any dimension, not just multiples of $n$, so that $p^*(K) = [L:K]\: p^*(L)$. However, I don't know how to prove the existence of such a basis in a $K$-agnostic way.
EDIT: A possible way to restate the problem is by noticing that for any $(v,w)$ in the direct sum $V \oplus W$, we have $\|(v,w)\|^p_p = \|v\|^p_p + \|w\|^p_p$, i.e. the $p$th power of $\|\cdot\|_p$ acts additively on "independent" vectors (for finite $p$, of course).
Given a length function on vector spaces $V$, we can define the associated Gaussian function as an integrable function $g : V \to \mathbb{R}_{\ge 0}$ depending only on length (i.e. factorizing as the composition $f \circ \|\cdot\|_p : V \to \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ for some $f$), and satisfying the independence property
$$g( (v,w) ) = g(v) g(w).$$
With the above choice of maximally symmetric length functions, using the aforementioned additivity property immediately leads to the usual Gaussian function $\exp(-k \|x\|_2^2)$ for $\mathbb{R}$ and to $\exp(-k' \|x\|_1)$ for $\mathbb{C}$ (where the constants $k, k'>0$ depend on the normalization convention). For $p=\infty$, this reasoning can't be used directly, but if we interpret it as a limit like before, we do recover the standard Gaussian function for non-Archimedean fields
$$\lim_{p\to \infty} \exp(-k'' \|x\|_p^p) = \begin{cases} 1 & \|x\|_p \le 1 \\ 0 & \text{otherwise} \end{cases} = \mathbf{1}_\mathcal{O_K^n}(x),$$
which is the indicator function of $\mathcal{O}_K^n$ in $V=K^n$. Focusing on the $1$-dimensional case $V=K$, these Gaussian functions coincide with the standard ones as used e.g. in Tate's thesis.
So if I'm not mistaken, the problem can be restated as proving that $[\bar{K} : K] = p$ if and only if the standard Gaussian function associated to $K$ is $\exp(-k \|x\|_p^p)$ (with a limit intended if $p=\infty$). A possible advantage of this restatement is that standard Gaussian functions have different characterizations that do not previously assume any length function (for example, they are their own Fourier transform, or satisfy analogues of the central limit theorem, see e.g. here).
$(\dagger)$ Note that by the Artin-Schreier theorem, the only degrees $[\bar{K}:K]$ that could possibly occur for any field $K$ are $1, 2, \infty$ (this, in turn, seems to be related to the fact that a nonzero integer can only have multiplicative order $1, 2$ or $\infty$, but that is a topic for another question).
$(\ddagger)$ As a fun fact, this can be shown to imply that the hypervolume of the $n$-dimensional complex unit ball $|x_1|+|x_2|+\ldots+|x_n| = 1$ , i.e. the ordinary $2n$-dimensional unit ball, equals $V(B_{2n})=V(B_{2})^n /n!=\pi^n /n!$; compare with the hypervolume $V(X_{n})=V(X_{1})^n /n!=2^n /n!$ of the $n$-dimensional cross-polytope$|x_1|+|x_2|+\ldots+|x_n| = 1$, where $| \cdot |$ now denotes the real absolute value.