Your code you added in OnInitializedAsync()

Input.FirstName = user.FirstName;
Input.LastName = user.LastName;

Should include null-coalescing operator like so:

Input.FirstName ??= user.FirstName;
Input.LastName ??= user.LastName;

Each time the page is loaded the OnInitializeAsync method is run even when a user hits Save so your old user values are loaded into the InputModel and what ever was changed in you form is lost unless you use the ??= which only replaces the InputModel's value if it is null.

Read more here null-coalescing operators