Your reasoning is sound. Larger frames raise the top tube and make it longer. It looks like you've already chosen a pretty short reach stem, and you're using swept back bars. Some people have back pain from such an upright position, but if it's comfortable for you and not causing any problems there's no reason to change it. If you're at the height limit of the stem you're using, a larger frame will let you put the bars even higher. The effect that has will vary from frame to frame. Different manufacturers have different sizing philosophy. Some increase the top tube length proportionally to the seat tube size, but some don't.

If you have medical insurance, it might be worth seeing if your insurance will cover a bit fit.