A Public Test Realm might not be in the budget of Anet's staff, plus it would require resources which everyone knows Anet doesn't have. They already have a very good, sound system (that in my opinion is superior to something like WoW's PTR system): The beta system.
A few weeks before major patches (maybe not for smaller patches), let us create test characters the same way beta characters work. They can't earn anything, they are snapshot characters, except using the new patch update so players that want to give input, the experts, the casuals, everyone can test for themselves and provide input here on the forums and Reddit. A day or two would never hurt and would fill the void Anet needs but can't provide.
One downside of a PTR is you need an abundance of players or you can't do things like raid testing, major group content, etc for real world testing. With beta characters, that's all fixed because they're on real servers.