On a plot of log(cases), per capita is just an offset. It shifts the line up or down on the plot, and doesn't make a difference. In a post upthread, mzs reminds us of the math:

log(cases/pop) = log(cases) - log(pop) = log(cases) + C

Log plots tend to visually compress relatively small effects such as these, so you don't really notice it, but it is much of why the lines aren't completely on top of one another.

In other words - a per-capita plot should look the same, but the lines would be harder to distinguish, because they'd be mostly right on top of one another.

You are assuming people are only interested in the slope. Then the constant doesn't matter. But it does matter when you want to know which country is an infection hot spot or not. A thousand cases in the US is something completely different from a thousand cases in Vatican city.