This basically sounds wrong to me. If you want accurate information, you'd be better off looking at books rather than videos on youtube. It's a little difficult to tell for sure whether this person is confused or is just not expressing themselves with sufficient precision. Again, that's one of the advantages of written communication: precision.
Full space is needed in GR.
This is already pretty weird. Are these the speaker's actual words, or is this a paraphrase or translation from another language? I don't think any relativist would refer to "full space" in contradistinction to Minkowski space. Even with the context, it isn't clear what this person means here.
However, Minkowski space is sufficient to compute properties of free falling bodies.
This doesn't really make much sense. Unless there's some missing context, they haven't described what they mean by "properties." A single pointlike, neutral, free-falling particle doesn't really have any interesting properties. It has some mass, and it's at rest in its own inertial (free-falling) frame. Check back later -- it still has that mass, and it's still at rest relative to itself.
To make this a nontrivial statement, you would have to have, say, two such particles. And then this statement becomes clearly false. For example, two such particles can follow geodesics that coincide at event A and then coincide again later at event B. That would be impossible in Minkowski space.
SR is sufficient to do frame transformations between [...]
This sounds like the lecturer doesn't understand that frames in GR are local, so, e.g., we can't have a frame that encompasses both a and b from their list. Or they may mean something different, but again it's impossible to tell because they aren't speaking precisely.
Related: How do frames of reference work in general relativity, and are they described by coordinate systems?