Another approach is to use a Workflow Validator. I think they can be used for new issues, before they transition to the first state in the workflow.
On that transition in the workflow, put a set of "Field required" validators, one for each field.
I believe it also allows you to include an error message for the user when the validation is not met. In a case like this, I'd suggest a message like "The Epic Link, Sprint and Original Estimate fields are required".
The decision to use field configuration or workflow validation might be influenced by which of those are shared across multiple projects, and whether this limitation applies to them all.
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