Should the sun appear very low in the horizon if its path is always between the two tropics?
Your explanation is correct, so you are not remembering correctly. The farther north you go, the lower the sun is to the horizon at noon.

Of course it depends on how far north you go. If you only travel 5° farther north in latitude, then the sun is only 5° lower in the sky. 5° is a "long" ways to travel on the earth, but in the sky 5° lower is not that noticeable.