I don't think PCB assembly would be that cheap, especially for a board with lots of SMT components. What I understand is that SMT manufacturing has a high startup cost, you need to supply the factory with full tapes of parts, and the factory needs to program the assembly line, often a factory is reluctant to take any offer below tens thousand boards because its high overhead. Even for service providers specialized in prototyping then, I don't see many provide QFN and BGA packages as options.
What have I missed here?
I just requested some quotes for assembly of 200 boards with a single-side load of 25 SMT parts, of which one is a BGA. The cheapest quote was ALLPCB at $1.09 per board, followed by PCBWay at $1.33 per board. That price includes setup and testing.
No transaction involving Chinese outsourcing is without risk and you need to do your homework, but it is no longer the case that DIY assembly is your only option.