The honest answer is that this isn't a binary. Power and heart rate measure different things, and pretending one replaces the other is one of the costliest mistakes amateur cyclists make. Power is mechanical output. Heart rate is the cost of producing that output. They're complementary inputs, not competing ones.
Power should be your primary target on hard days. It responds instantly, doesn't drift with caffeine or temperature, and is repeatable across rides. When you're prescribed 4×8 minutes at 105% FTP, the power number is what you hit. Heart rate would arrive 60-90 seconds late and would be elevated by heat, dehydration, or under-recovery — exactly the variables you're trying to ignore for a structured interval session.
Heart rate earns its keep on long endurance rides and as a fatigue/recovery signal. On a 4-hour zone 2 ride, your heart rate at the same power tells you whether you're well-rested or quietly cooked. A drift of 10+ bpm at constant power 90 minutes in (cardiac drift) is normal; a drift of 20+ bpm screams under-recovery, dehydration, or under-fuelling. The Roadman Comparison page on heart rate vs power has the full breakdown of when each metric leads.
Practically, the elite amateurs Anthony has interviewed all use the same hierarchy. Hard sessions: power leads, HR is the witness. Endurance/recovery sessions: HR leads (so you don't accidentally ride too hard), power is the witness. Race day: power for pacing, HR for spotting trouble before it becomes a bonk. That's the pattern Joe Friel and the Roadman coaching team apply, and it's the one most coached amateurs end up using whether they realise it or not.