2004 Audi S4 Question cold weather starting problem

I'm having an odd cold weather starting problem with my 5yr old S4, which recently developed along with winter. When starting the car 1st thing in the morning: I turn the key, all lights come on as normal, turn key to start and nothing happens. No warning lights, no whine from a failing starter motor. Absolutely nothing.

Keep the key turned and within roughly 10 seconds it starts as normal. But I never know if it'll be 2 seconds or 20. Worst was 4 minutes of starting attempts. It seems that the colder it is out, the harder it is to start.

However, once warmed up, regardless of outside temp, it turns over immediately. Seems like anything less than 8 hours of unused time, it starts fine.

Dealership is saying that it's likely the starter motor (for which they want $1,000 to install!), But it sounds like the starter motor is working perfectly, once engaged. It's not whining or clicking. It sounds like more of an electrical issue. Thoughts? Thank you!