Here is a hypothetical.
My guess it that something is caught in its throat. Maybe a piece of junk with some plastic grocery bag on it. When it gets a full flush, the stuff gets jammed up and blocks the water. When you pour in some water slowly, it dribbles through.
Test it while you have it off the system. Pour some water in the tank and flush it. See if you get the same result.
I would do a ramrod cleanout from the outlet end. Maybe you could get a line through it and pull a something large back and forth through it to be sure it is really cleaned out.
If the toilet works when it is disconnected but not when installed, maybe it is a problem with the plumbing and there could be a problem with a new one.
Just thought of another possibility. If there is a leak at the high point in the passageway it could cause loss of the siphon effect and make it flush poorly.