to watch me play - what rules are here

Question

Here is an excerpt from The Art of Learning:

More and more fans came to my tournaments to watch me play and get autographs.

My questions is should not it be:

1. either More and more fans came to my tournaments to watch my games and get autographs
2. or More and more fans came to my tournaments to watch me playing and get autographs
3. or, in order to avoid fused gerund, More and more fans came to my tournaments to watch my playing and get autographs, but to be honest this option sound weird to me and it is better to use the first option.

If me play is fine, then what are the rules, can you delineate a syntax tree for the sentence?

P.S. I am primarily interested in BrE, if it matters here.

Answer 1

As a native English speaker, the sentence sounds fine to me.

If it helps, I see the the base sentence as "They came to watch me". Almost any verb can be added after "me". They came to watch me sing, or watch me dance.