This post/question is admittedly vague, but I hope that with some feedback in comments it could be made more precise.

For $E$ a Banach space, $K(E)$ and $B(E)$ will denote the Banach algebras of compact and bounded linear operators on $E$, respectively. There is a tradition / industry in the world of Banach algebras that asks whether $K(E)$ or $B(E)$ can display various properties if we choose $E$ judiciously. As part of this we would like to understand these algebras when $E$ is one of the "classical" Banach spaces such as $\ell_p$ or $L_p$ (with the convention that the unadorned $L_p$ is an abbreviation for $L_p[0,1]$).

On the other hand, we have made progress in establishing "bad" behaviour of $B(E)$ when $E$ is a space such as $\ell_p\oplus \ell_q$ when $p\neq q$, or various $\ell_p$-sums of $\ell_{k(n)}^n$ ($n\in {\bf N}$), not to mention "exotic" examples that have been constructed for the explicit purpose of giving $B(E)$ certain properties.

Here are my questions.

Q1. Is there any way to justify, with mathematical definitions or results, the feeling that $\ell_p$ and $L_p$ are more "symmetrical" examples than the various other examples mentioned above? Note that in various senses $\ell_p\oplus\ell_q$ is better behaved than $L_1$, e.g. the latter space does not have an unconditional basis, while the former space has a very nice unconditional basis.

Q2. $\ell_p\oplus\ell_q$ and the $\ell_p$-sums of finite-dimensional $\ell_k$ have a flavour of "gluing together things which look different", and hence one might argue that it is not so striking that for such $E$ the algebra $B(E)$ has behaviour very unlike $B(\ell_2)$. Are there established notions in the Banach-space literature that capture the way $L_p$ and $\ell_p$ look much more "homogeneous"? (The phrase "homogeneous" is meant in a loose sense, not in the sense of https://arxiv.org/abs/math/9205207 )

Regarding Q2: I know that $\ell_p$ is a prime Banach space for each $p$ (every complemented subspace is isomorphic to the whole space) and that $L_p$ is primary (in any direct sum decomposition of the space, at least one of the summands is isomorphic to the whole space), but my understanding is that primeness is too restrictive while being primary allows many more examples. So candidates for answers to Q2 would be notions that are intermediate between "being prime" and "being primary", if such exist.