What are the downsides of publishing your statement of purpose on the Internet after you've already been admitted to a PhD program?
In general, I think this sounds like a great idea.  I would further say that the same essential content is often a good starting point for your personal professional website.

The only potential downside that I see is that in the future you might look back with embarrassment on naive ideas in your statement where your feelings have evolved over time.  If you have enough self-confidence to be OK with with that, it's fine: nobody will expect a mature researcher to believe all of the same things they did when just starting grad school.