```python
def apply_edit_script(original: str, script: list) -> str:
    if not isinstance(original, str):
        raise ValueError("Original must be a string")
    
    result = ""
    i = 0
    
    for op in script:
        if len(op) != 2:
            raise ValueError(f"Invalid op {op}")
        
        op_name, count = op
        
        if op_name == "copy":
            if not isinstance(count, int) or count <= 0:
                raise ValueError("Copy count must be a positive integer")
            result += original[i:i+count]
            i += count
        elif op_name == "skip":
            if not isinstance(count, int) or count < 0:
                raise ValueError("Skip count must be a non-negative integer")
            i += count
        elif op_name == "insert":
            if not isinstance(count, int) or count <= 0:
                raise ValueError("Insert count must be a positive integer")
            text = original[i:i+count]
            result += text
            i += count
        else:
            raise ValueError(f"Unknown op {op}")
    
    if i != len(original):
        raise ValueError("Script stops before the original is used up")
    
    return result
```