Why are quasi-isomorphisms of homotopy algebras only defined for arity 1?

Q1: The conventional theory of homotopy algebras is built on the premise that the lower-degree operations dominate over the higher-degree ones, in some sense. A discussion of this can be found in the introduction to my preprint "Weakly curved $\mathrm A_\infty$-algebras over a topological local ring", http://arxiv.org/abs/1202.2697. This does not answer your question fully, but explains the underlying ideology to some extent.

Q2: The theory of curved DG-algebras/curved DG-coalgebras and the co/contraderived categories of CDG-modules/comodules/contramodules over them is built on the premise that the (co)multiplication dominates over the differential (and the differential dominates over the curvature). So the higher-degree operations dominate over the lower-degree ones in these "theories of the second kind". On CDG-coalgebras or DG-coalgebras, there is even a model structure with such weak equivalences (though a precise definition is more complicated and maybe does not accord to what you describe in the question). See my memoir "Two kinds of derived categories, ...", http://arxiv.org/abs/0905.2621.

Q3: It is indeed true that the homotopy category of DG Lie algebras is equivalent to the homotopy category of $\mathrm L_\infty$-algebras, though perhaps for reasons more complicated than described in the question. Similarly, the homotopy category of associative DG-algebras is equivalent to the homotopy category of $\mathrm A_\infty$-algebras.

Basically, with any $\mathrm A_\infty$-algebra one can naturally associate a much bigger DG-algebra quasi-isomorphic to it; while for any DG-algebra one can, if one wishes, construct a (generally speaking) much smaller $\mathrm A_\infty$-algebra quasi-isomorphic to it (in addition to the obvious option of viewing a DG-algebra as an $\mathrm A_\infty$-algebra with vanishing higher operations).

But of course, one cannot just forget the higher operations of an $\mathrm A_\infty$-algebra and obtain a quasi-isomorphic DG-algebra, firstly because the multiplication in an $\mathrm A_\infty$-algebra need not be associative, and secondly, even if it is, the identity map cannot be extended to an $\mathrm A_\infty$-morphism (in most cases).

My favorite explanatory analogy for the first question, along the line of Leonid's answer, is that a power series in one variable with no constant term, $a_1x +a_2x^2 +\cdots$ has an inverse under composition if and only if $a_1$ has an inverse in the ground ring.

Here is a down to earth explanation, by way of analogy.

A group homomorphism $\varphi:G\to H$ is an isomorphism of groups iff it is a bijection of sets. Why do we not need to say anything about the group operations coinciding? The reason is simply because the fact that $\varphi$ is a group homomorphism already implies this.

There is an essentially endless list of similar situations, and quasi-isomorphisms of $A_\infty$-algebras is one of them. Of course, the fact that quasi-isomorphisms of $A_\infty$-algebras (over a field!) are invertible up to homotopy is by no means automatic, but it is straightforward to check.