You've already done more diagnostics than most people, and based on what you've listed I'd be reluctant to condemn the ECU at this stage.
If the ECU relay circuit was genuinely losing power, I'd expect stalling, resets, communication faults, or a no-start condition rather than a repeatable hunting idle.
The relay buzzing is interesting, but relay chatter is often the result of a voltage drop rather than the cause. I'd want to know exactly what voltage is present at relay 100 terminal 30, terminal 87, and the ECU supply pins when the fault actually occurs.
I'd also pay close attention to the fuse carrier on top of the battery. (If it has one) Mk5/Mk6 platform cars are well known for corrosion, heat damage and high resistance connections in that area. A visual inspection isn't always enough; voltage-drop testing under load is often more revealing.
As for the hunting idle itself, the timing coincides almost perfectly with the ECU transitioning away from cold enrichment and into closed-loop control. That makes me wonder whether there is a separate engine-management issue causing the unstable idle, with P0687 being a secondary consequence.
Things I'd check next:
* Live data from cold start until the fault appears.
* Lambda sensor readings before and after the idle starts hunting.
* Short-term and long-term fuel trims.
* MAP sensor readings at idle.
* Coolant temperature sensor plausibility.
* Crankcase ventilation (PCV) system and vacuum leaks.
* Throttle body adaptation status.
Regarding your questions:
1. Yes, I've seen P0687 logged where the relay itself was fine and the root cause was wiring resistance, poor power supply, fuse carrier issues or voltage-drop problems.
2. Wiring faults are far less common than power distribution and fuse carrier issues, but they're possible, especially where moisture has been present.
3. Yes. If it were my car, the battery-top fuse carrier and associated power feeds would be near the top of my list.
4. An ECU driver fault is possible but relatively rare. Given that the ECU appears to control the engine normally during startup, I'd treat the ECU as the last suspect unless power and ground integrity have been proven good.
5. Definitely check and voltage-drop test all ECU and engine earths rather than simply inspecting them. An earth can look perfect and still have enough resistance to cause intermittent control circuit faults.
Personally, before spending money on an ECU, I'd get VCDS logging on ECU voltage supply, coolant temperature, fuel trims, lambda control and throttle angle from cold start until the idle begins to hunt. I suspect the transition into closed-loop operation may be the key to solving this one.
