Pyretic Ritual is not a mana ability and doesn’t behave like one. You cast it as a spell. It uses the stack. It can be responded to, and it can be countered.

In the same vein, you can cast Pyretic Ritual only at a time when you could cast any other instant, not at a time when you could activate a mana ability. For example, you can’t cast Pyretic Ritual in the midst of casting another spell; you’d need to cast Pyretic Ritual first.