The Tube Does Not Run Early Enough
The London Underground does not run 24 hours on most lines. The Piccadilly Line to Heathrow — the direct rail link — runs from approximately 5:02am on weekdays from central London stations. For a 6am flight you need to check in by 4:30am, meaning you need to leave Central London around 3:30am. The Tube simply does not exist at that time.
Even on Fridays and Saturdays when limited night services run, the Piccadilly Line to Heathrow is not part of the Night Tube network. A chauffeur is the only public-facing transport option for very early morning Heathrow departures.