If neither the camera, nor the time are changing, then ofc it'll look like it's frozen (because it is).
Try the following:
You can keep your Position keyframes, a camera that's not moving is ok.
Now, remove both of your time keyframes.
Travel to the time at which you start building. That is, where you want your video to start.
Then place your first time keyframe (at 0:00 on the lower timeline) while at that time in the replay.
Now travel in time to the point at which you're doing recording and place your second time keyframe (at 3:00 on the lower timeline, or however long you want your video to be).
Finally, render your video. You might want to make a test render with minimal settings (10fps, 300x200 res, potato quality) first to avoid wasting another two hours.