I think a reasonable partial explanation comes from universal algebra. The lattice Con(A) of congruences of an algebra is always a complete algebraic lattice. Therefore, it is meet continuous in the sense that $\bigvee_i a \wedge b_i = a \wedge \bigvee_i b_i$ whenever the $b_i$ form a directed family of congruences. When Con(A) happens to be finitely distributive, then one can drop the 'directed' requirement. In this case, Con(A) becomes a frame and it can thus be viewed as an abstract topological space (i.e. a locale). In fact, since Con(A) is algebraic the corresponding locale is always spatial and it always corresponds to a concrete spectral space.

In the case of a ring A, the lattice Con(A) is isomorphic with the lattice Id(A) of ideals of A. The lattice Id(A) is not always distributive. (Though it is when A is a Prüfer domain and hence when A is a Dedekind domain, for example.) To remedy this, one looks at the radical ideals of A, which are always better behaved, to define the Zariski spectrum. In my humble opinion, the existence of radicals makes rings very special among algebras.

I think a reasonable partial explanation comes from universal algebra. The lattice Con(A) of congruences of an algebra is always a complete algebraic lattice. Therefore, it is meet continuous in the sense that $\bigvee_i a \wedge b_i = a \wedge \bigvee_i b_i$ whenever the $b_i$ form a directed family of congruences. When Con(A) happens to be finitely distributive, then one can drop the 'directed' requirement. In this case, Con(A) becomes a frame and it can thus be viewed as an abstract topological space (i.e. a locale). In fact, since Con(A) is algebraic the corresponding locale is always spatial and it always corresponds to a concrete spectral space.

In the case of a commutative ring A, the lattice Con(A) is isomorphic with the lattice Id(A) of ideals of A. The lattice Id(A) is not always distributive. (Though it is when A is a Prüfer domain and hence when A is a Dedekind domain, for example.) To remedy this, one looks at the radical ideals of A, which are always better behaved, to define the Zariski spectrum. In my humble opinion, the existence of radicals makes commutative rings very special among algebras.