It seems that specifying a value (value=) as an attribute of a calcite component such as calcite-input or calcite-select does not make that value immediately retrievable through the calcite element's element.value
https://codepen.io/David-Jantz/pen/QwWprgy
Programmatically, in JS, you could get the calcite element's 'value attribute' via element.getAttribute("value") and subsequently assign the 'value' attribute's value to the element's value with:
element.value = element.getAttribute("value"). However, it's unclear why this step is necessary when components like calcite-input already contain events for updating the element.value when a user types it as an input. If a calcite component allows the developer to specify the "value=" attribute already, then what other purpose would that have than being the desired default element.value? It seems effectively no different than a placeholder.