If the Director happened to be fired while overseeing an investigation that could directly or indirectly implicate the President with the intent to end or to slow such an investigation, there could be a serious problem.
You’ve phrased the matter accurately, which very few do.
Most want to replace the word “problem” with “crime” which is bunk.
The remedy for a President who has acted in what normally would be considered criminal manner is impeachment, not indictment. And the people who would dispute that “crime” is inaccurate should sit back and count their blessings; an impeachment is carried out in the court of public opinion, and the standard will be the preponderance of evidence.
If this had to proceed to a criminal trial, the standard would be beyond a reasonable doubt, and proving that would be nigh on to impossible, since you have to prove intent.