I think they *would* keep drawing more and more rights from a weak Chinese government. That's why they didn't try to conquer China in OTL, even though it was within their power to do so.It also took the British in India a couple of hundred years to achieve their end, and was taking over not one state but a myriad of smaller ones which is an order of magnitude more easier than taking over a unitary state. And even then they left a bunch of pseudo-independent princely states rather than outright annexing the whole thing.
"Method of conquest" has everything to do with it, because it determines the logic behind the conquest in the first place. Why would you go around annexing Chinese territory? What is the logic behind this and why is this more profitable than simply continuing OTL's policy of using a weak Chinese government to further western interests?
I was simply speaking to what would likely happen if they did attempt it, as per the OP.