I agree that none of the options mentioned so far seem right, even if we could theoretically construct explanations for them.
'Would get to play' would be a strange meaning, similar to 'would be allowed to play a role' or 'would be able to play a role.' That seems logically unlikely. 'Get to play' is usually seen in other contexts, e.g., 'He got to play when the first-string player was injured.'
'Might have played' also seems logically unlikely, since the sentence doesn't continue with anything contrary to fact. (E.g., '..., but it was not to be / ..., if he had not been killed at a young age / ..., if our paths in life had continued as we then expected them to continue.'
Similarly, 'has had to play' is not grammatically impossible -- it's not actually the past tense, but the present perfect -- but it too is logically unlikely, because 'to have to play a role in someone's life' is an unlikely situation. ('The boy I was about to meet has had to play an important role in my life. When I became an orphan, his parents adopted me, and since then he has had to accept me as his sibling whether he likes it or not.')
To me the obvious correct answers, one of which you would expect to see among those offered, are as follows:
The boy I was about to meet _______ a very important role in my life.
was to play
would come to play
This is what I think of as 'future in the past,' whether or not that's the right formal name for it.
I think I know, or knew, it in Romance languages, but I'm not sure we ever covered it in German class, so I've never actually been quite sure what corresponds to it, if anything, in German. A recent thread where it came up never really reached a definitive conclusion, unless dude's 'sollte' was it and it just never was confirmed by anyone else.