There seem to be two main points here.
The first is is that ‘He will come to pick you up’ is a very unconvincing specific example. I cannot imagine anyone ever saying that, while I can imagine people saying, ‘He will come to fix your boiler,’ or, ‘He will come to steal your children.’
In philosophical analysis mode I would say that, in the given example, coming and picking up are essentially aspects of the same action. That is, you don’t come to pick someone up: in coming, the picking-up is achieved. In the case of fixing your boiler or stealing your children, coming is a means to the end of then doing something else.
Secondly, this also appears to rest partly on a difference between common usages in British English and American. ‘He will come pick you up’ sounds American to me (an experienced speaker of British, and an experienced viewer of American TV).
In the UK it is vastly more likely—in fact I would say it would be normal—for someone to say, ‘He will come and pick you up.’ I have looked around for a while this morning, and can find no grammatical justification for that construction. I hope that my compatriots might at least confirm the very common usage.
In fact we might just as easily and correctly say, ‘He will come and fix your boiler,’ or, ‘He will come and steal your children.’ In the coming, the further result is made at least possible, and perhaps inevitable.
It is also possible that this example might also have suffered accidental contamination from expressions like ‘He will come to regret this,’ meaning that a time will arrive when he understands that this was a bad idea all along. I don't know why I chose a negative example. I might equally have said, ‘He will come to appreciate this.’
In the end, when the guy actually turns up, the important thing might be simply to check whether he is a taxi driver, a plumber or a demon.