OK, I would challenge you to find a single case, reported in the media, in which a defendant took a plea deal rather than face charges that included mandatory sex offender registration for nothing more than public urination. If that has ever happened, surely it would've been reported on at least once, right?

I am uniquely qualified to respond to your question in that over my career I have represented 4,000+ criminal defendants. It is incredibly common for defendants to take lower charges to avoid consequences. This is nearly uniformly true for sex offenses which require registration. While obviously not able to use individual client names, I can guarantee I have personally represented hundreds of defendants who pled to a felony obscenity charge to avoid to avoid registration as a sex offender.

And were any of those for public urination alone, without any obscene intent at all? Just a guy peeing in the bushes because he can't find a bathroom?

My response was to the question about defendants pleading out to a lesser charges to avoid felonies. But, I have pled several defendants to charges based on public urination, both guilty as charged (typically to avoid a Habitual Offender enhancement) and to misdemeanor charges. In my state, Obscenity (which public urination is considered) is not a registration requiring offense, but it is in several other states, like Texas for example.