Knot diagrams are a special sort of tangle diagrams, so I will reinterpret your question as being about tangle diagrams. Tangle diagrams are a "planar algebra" generated by ${\text{overcrossing},\text{undercrossing}}$, so every tangle can be drawn by taking a finite collection of generators, arranging them in a plane, and connecting each of the four "loose ends" on each generator by "bridge arcs" to another of the "loose ends" (of the same or a different generator), or leaving "the loose end" "loose". The relations are Reidemeister relations. If you allow "bridge arcs" to cross (and allow virtual Reidemeister moves), you get virtual tangles, and if not, you get usual tangles.
This is already algebra, but it's algebra in a different sense from "x+3=2" because it takes place in the plane. You could introduce a height function and translate tangles into "algebra" in the old sense, as some other answers suggest, but surely to do so would constitute an act of violence. Maybe it's better (philosophically at least) to widen one's perspective on what constitutes "algebra".
I certainly think that yes, "there are important theorems with no known proof except via diagrams, and nobody knows why". Anything proven by using skein relations fits the bill. Nobody really knows what quantum invariants have to do with 3d topology (other than the Alexander polynomial for links, but the tangle version of the Alexander polynomial also fits the bill), but it's quite clear what they have to do with diagram algebras if they are defined via linear skein relations.
Surely more than that is true- many invariants of knots extend naturally to invariants of more general "diagrammatic algebras", and maybe this wider context is where we can understand those invariants and where they make more conceptual sense. Maybe coming to terms with "the metamathematics of diagrams" (tangle diagrams, and more general classes of diagrams as well) as a brave new algebra is a fruitful direction of research. I interpret current work of Dror Bar-Natan in this vein.
As a concrete example of where this concept has proven useful, see Zsuzsanna Dancso's thesis, which (building on ideas of Bar-Natan and D. Thurston) explains how considering diagrams of knotted trivalent graphs (a larger "brave new algebra") helps us to understand how the Kontsevich invariant of a framed link changes under handle slides (Kirby 2 moves). Even more so, Bar-Natan and Dancso's forthcoming w-knotted objects project is an example of a setting in which taking "the metamathematics of diagrams" seriously, treating them with respect as a genuine form of algebra, motivates the project and yields substantial dividends, at least in the form of better understanding the Alexander polynomial of tangles.
