Test functionality with timeouts in react-testing-library and jest

I'm trying to follow TDD, and I have a span that should appear on screen after 5 seconds. I haven't implemented the span at all, so the test should fail, but currently it passes the test expect(messageSpan).toBeInTheDocument.

Here are my two tests:

it("doesn't show cta message at first", () => {
    render(<FAB />);
    const messageSpan = screen.queryByText(
      "Considering a career in nursing? Join our team!"
    );
    expect(messageSpan).toBeNull(); // passes, as it should
  });

  it("should show the cta message after 5 secs", () => {
    render(<FAB />);
    setTimeout(() => {
      const messageSpan = screen.getByText( // also tried queryByText instead of get
        "Considering a career in nursing? Join our team!"
      );
      expect(messageSpan).toBeInTheDocument(); // also passes, even though messageSpan should throw an error.
    }, 5000);
  });

Here's my FAB component, where you can see there's no message at all:

export default function FAB() {
  return (
// using styled-components; there's no content in any of these.
    <StyledFABContainer>
      <StyledFABButton>
        <BriefcaseIcon />
      </StyledFABButton>
    </StyledFABContainer>
  );
}

To complicate things, I don't plan to have a set function I call for the setTimeout. I will simply set state after a set time of 5 secs. So don't think I can use the suggestions in the timer mocks section of jest docs: https://jestjs.io/docs/timer-mocks

My two questions are:

a) Why would this pass and not throw an error/null? b) How can I properly test setTimeout functionality in RTL?

UPDATE: Have tried using various combinations of useFakeTimers, act, waitFor etc., but no luck. Here's my current test as it's written out, and throwing two errors - one saying I need to use act when changing state (which I am, but still) and one saying my messageSpan is null:

it("cta message to display after 5 secs", async () => {
    jest.useFakeTimers();
    const el = document.createElement("div");
    act(() => {
      ReactDOM.render(<FAB />, el);
    });
    jest.advanceTimersByTime(5000);
    const messageSpan = screen.queryByText(
      "Considering a career in nursing? Join our team!"
    );
    expect(messageSpan).toBeInTheDocument();
  });