The raising of blood sugar from protein is demand driven. But the conversion of protein into carbohydrate is also supply driven.
We catabolize excess protein. How that happens depends on the specific amino acids. Several, for example glycine, are broken down into pyruvate. If ATP is low pyruvate from amino acids is oxidized exactly the same as pyruvate from glucose. If ATP is high pyruvate is converted to glucose.
There are other gluconeogenic (glucose precursors) breakdown products of amino acids such as α-ketoglutarate, succinyl-CoA, fumarate, and oxaloacetate. There are also a couple ketogenic breakdown products acetyl-CoA and acetoacetate and the net effect of consuming protein is roughly half as anti-ketogenic as consuming carbs.