Do these new pieces solve the Cold Start Issue?
Or the problems are not related?
No, they are not directly related. Indirectly they might be slightly related. The fix is refering to a charging issue. If you can't get your battery charged, you can't get your engine started in any weather, cold or hot ;-)
Move one step up on the distribution ladder. Contact the guys above your service workshop. Be prepared to show your service history and that the problem hasn't been solved in spite of many tries. That's what I would do.