Personally, I first go online to see whether I can find a solution or more info about a problem, but I think that is an approach typical for the more technically inclined types. 'run of the mill' consumers, I think, will directly call the shop where tey bought stuff. I do not think they will call whoever made the phone, if only because they often do not know how to do that.
Yes, there are cases where one can register a product with the producer and get a guarantee that way, but even then, EU law is clear: the buck stops at whoever sold you the device.
So, even then, I would be inclined to contact the seller earlier than the producer because, in the end, if you buy form Q, Q is the only one who has an obligation to help you. In this case, If I were to buy from Google, and things didn't work as expected, I would contact Google. They cannot hide behind "but we bought it from X". If they could, consumers would typically get no support at all. For example, X would say "it looks like the battery is the problem. That is not ours; we bought it from Y"; Y would say "I don't know. Maybe it's the controller chip? Z made that"; Z would say "we outsourced the software for that chip" or "no, it's not the chip, it must be the chemicals in the battery/the device getting too hot due to incorrect firmware in some other CPU", etc.