Because in Europe, if you do what you say you want to do, the government pays off child support for you to the mother, then turns around and tries to collect from you, including getting a default judgement if you decide to be a no-show.
So you're not messing with her anymore, you're messing with an European government. Once you're messing with a government, reciprocity among different European nations applies.
On top of that, read this carefully:
United States and Agreements with Other Countries
The United States has reciprocal agreements with several foreign countries for the enforcement of child support orders through the Secretary of State. Currently, the United States has treaties with Australia, Canada, Czech Republic, Ireland, Netherlands, Norway, Poland, Portugal and Slovak Republic regarding the enforcement of child support orders. The United States also has similar agreements with Austria, El Salvador, Finland, France, Germany, Hungary, Ireland, Israel, Sweden, Switzerland and the United Kingdom.
The Office of Child Support Enforcement also assists states and countries with the enforcement of international child support orders. It assists the United States Department of State with negotiating and implementing foreign reciprocating countries and provinces and the most recent international Hague treaty for child support. The United States currently has 26 such foreign reciprocating countries and provinces that it has arrangements with.
That's a lot more countries than what you listed, and pretty much any important country in Europe save for Italy and Spain. Again, I don't think you did your homework on this.