Under which metabolic circumstance would amino acids be routed toward gluconeogenesis rather than toward new protein synthesis?

Whenever dietary fat intake is above average, because high fat intake prevents any amino acid from being used in anabolic pathways including glucose production.

Immediately after a high-protein meal when insulin and amino acid availability drive exclusive and complete conversion of amino acids into lipid storage rather than glucose or proteins.

During the first 5 minutes after consuming a carbohydrate-rich snack when gluconeogenesis is maximized to clear excess blood glucose.

During prolonged fasting or very low carbohydrate intake, when maintaining blood glucose becomes necessary and amino acids are used as substrates for glucose production.