Usually, rather than triangulating algebraic varieties, it is better to find cylindrical decompositions. These are finite cellular decompositions that exist for every real semi-algebraic set and thus, forgetting scalars, for every complex variety. See Algorithms in Real Algebraic Geometry for more.

If you want a proof of finiteness of $H^{\ast}_{DR}$ without resolution of singularities and without topological tools, you can see Monsky, Finiteness of de Rham Cohomology. I wouldn't call it simple, though!

Usually, rather than triangulating algebraic varieties, it is better to find cylindrical decompositions. These are finite cellular decompositions that exist for every real semi-algebraic set and thus, restricting scalars, for every complex variety. See Algorithms in Real Algebraic Geometry for more.

If you want a proof of finiteness of $H^{\ast}_{DR}$ without resolution of singularities and without topological tools, you can see Monsky, Finiteness of de Rham Cohomology. I wouldn't call it simple, though!