CSIDH: Why do we need ideals in the form of $\langle \ell, \pi \pm 1 \rangle$ in order to apply Vélu's formulas when computing the action?
You are right that Vélu's formulas also work for any extension field, but then they require arithmetic in that field.
Doing computations in $\mathbb F_p$ is simply much more efficient than in extension fields.

Generally, for a "random" ideal of prime norm $\ell$, you can only expect the corresponding points to be defined over $\mathbb F_{p^k}$ for some $k\in\mathcal O(\ell)$, hence if CSIDH didn't have those rational points, one would have to compute in extension fields of degrees of a few hundred.

(In fact, the earlier variants¹ of CSIDH were, among other techniques, using subgroups that are not necessarily rational, because it seems hard to find an ordinary curve with many "good" subgroups. The core observation in CSIDH is that switching to supersingular curves makes this trivial, simply by picking the prime $p$ of a special form!)

¹ Couveignes, Rostovtsev–Stolbunov, De Feo–Kieffer–Smith