Ah, yes, the Paxman reference would have been much funnier, damn!
It feels like I've answered your question a dozen times already, but at the risk of boring everyone senseless here it is again.
I don't accept that your two choices are the only options.
If any of my kids (potty trained or not) could not hold their wee long enough to find a toilet. I'd make sure they wore a nappy when leaving the house until such times as they could.
In fact, looking back 20 odd years through the mists of time, I think that's exactly what happened. So the need to urinate in public never arose in my kids' case.
As the earlier poster said, we're never going to agree on this. So let's agree to differ.