Actually what's happening is a bit different. The one who observes the thinking arises in relation to what is thought ("the one who is thinking"--in your words

Actually what's happening is a bit different. The one who observes the thinking arises in relation to what is thought ("the one who is thinking"--in your words). This means that as the thought changes from one to the next, the one who observes the thinking also changes. The thought and it's observation arise together, and pass away together. This is followed by the arising of another (different) thought and corresponding observation of that thought. Prior to perceiving this inconstancy of observation/consciousness whatever consciousness that arises is identified as I, mine, that I'm within it, or that it's within me. When perceiving inconstancy of consciousness, whatever change of consciousness that arises is understood as just another change and is not identified with as me/mine/etc.