Sorry for taking so long to respond.
I agree with you that log-odds are hard to understand, but odds-ratio on the natural scale are also not great as the skew makes it hard to appreciate when the difference is actually symmetric. I.e. the plot can look quite differently if you choose to make the comparison in the opposite direction. I further think showing the full distribution is often not a great use of screen space, unless you are interested in the fine details. What I’ve found useful is to show posterior intervals of odds ratio on the log scale (i.e. the labels reflect odds ratio, but the scale is log-odds). Here is how it looks like for an analysis I’ve done, the plot is a bit crazy as this shows ALL THE COMPARISONS across seven genes for eight conditions, but I think it illustrates neatly the effectiveness of the representation:
The horizontal axis is the odds ratio on the log scale. Thin lines are 95% central posterior intervals, thick lines are 50% intervals. Color indicates the widest central posterior interval excluding 0 (would probably not be necessary if you have fewer comparisons). I particularly like how 95% interval just excluding 0 and 95% interval just including 0 look almost identical and that no sort of point estimate is visible, making it harder to overinterpret the results.
Here is a less crowded version showing odds ratios for each gene against the average for each condition:
The plot also shows odds ratios in groups of underlying data - those are the gray circles.
Does that help?