"Fool's covering spaces" are very close to *overlays* of R. H. Fox (see this paper in the first place and also this one), which I think are still better: they retain all nice properties of "fool's covering spaces" and have additional ones. An equivalent (see "Steenrod homotopy", Lemma 7.3 or Mardesic-Matijevic) definition of an overlay is that it is

a covering that is induced from some covering over a polyhedron (or equivalently from some covering over a locally connected semi-locally simply-connected space).

Fox's original (equivalent) definition is that it is

a map $p:Y\to X$ such that there exists a cover $\{U_\alpha\}$ of $X$ satisfying

(i) each $p^{-1}(U_\alpha)=\bigsqcup_\lambda U_\alpha^\lambda$, where each $p$ restricted over $U_\alpha^\lambda$ is a homeomorphism onto $U_\alpha$; and

(ii) if $U_\alpha^\lambda\cap U_\beta^\mu$ and $U_\alpha^\lambda\cap U_\beta^\nu$ are both nonempty, then $\mu=\nu$.

Condition (i) of course amounts to a definition of a covering in the usual sense.

A third definition of overlays is by their monodromy. $d$-Sheeted overlays over a connected base $X$ (possibly $d=\infty$) are identified with

the homotopy set $[X,BS_d]$.

This is essentially the monodromy classification theorem of Fox; for a shorter proof and the above formulation see "Steenrod homotopy", Theorem 7.4. Another reformulation:
overlays are

functors $pro$-$\Pi_1(X)\to Sets$, where $pro$-$\Pi_1$ is the fundamental pro-groupoid.

This is due to Hernandez-Paricio (but note that his claim that Fox did his theory only for finite-sheeted overlays is not only incorrect but misleading; in fact, for finite-sheeted ones Fox shows that they reduce to coverings). I'm not fully happy with the pro-groupoid definition because a pro-groupid is a whole diagram of groupoids. I would prefer something like "overlays are functors $\Pi_1\to Sets$, where $\Pi_1$ is the topologized Steenrod fundamental groupoid (which combines Steenrod $\pi_0$ and Steenrod $\pi_1$)"
Such formulation is possible, at least, in a special case (see Corollary 7.5. in "Steenrod homotopy").
Over a base that is compact and Steenrod-connected (aka "pointed 1-movable"; in particular, this includes compact spaces that are connected and locally connected), overlays are
identified with functors $\check\pi_1(X)\to Sets$, where $\check\pi_1$ is the topologized Cech (or Steenrod) fundamental group.
Note that $\check\pi_1(X)=\pi_1(X)$ if $X$ is locally connected and semi-locally simply-connected.

Finally, I should mention that over a compact (metric) base, overlays can also be defined
(Theorem 7.6 in "Steenrod homotopy") as

coverings in the category of uniform spaces.

Such uniform coverings have been studied by I. M. James in his book "Introduction to Uniform spaces"; see Brodsky-Dydak-Labuz-Mitra for a clarification of James' definition (the latter paper also has some relevant followups). This is really saying that overlays are precisely those coverings for which a metric on the base can be "lifted" to a metric in the total space. (Note that the compact base has a unique uniformity: as everyone might remember, every continuous function on a compact space is uniformly continuous.)

DISCLAIMER: Following Fox, I have been assuming all spaces to be metrizable :) It is known
that this is not a real restriction, and everything extends to arbitrary spaces, perhaps with minor modifications (see Mardesic-Matijevic's paper, which also has many additional references about overlays; also the papers by Dydak-et-al. and Hernandez-Paricio may be relevant to this point) However, I prefer being ignorant of the non-metrizable world and so don't follow these modifications or whether they are needed.

SUMMARY: For purposes of proving something about coverings of locally connected semi-locally simply-connected spaces usual covering work fine. For purposes of proving anything in topology beyond these restrictions, you would definitely need overlays, rather than "fool's covering spaces". But admittedly overlays are slightly harder to define. Thus for purposes of defining a formal concept which agrees with coverings for "nice" spaces and is not intended to be used for proving anything beyond "nice" spaces, "fool's covering spaces"
suit well; I would call them e.g. *path-overlays*.

By the way, I like the idea about the Seifert-van Kampen theorem; I think if combined with overlays, it should give a Seifert-van Kampen theorem in Steenrod homotopy, which would be an interesting result.