The ability to travel is not depending on visa.

Poland and Chech Republic are part of the EU, which is a difference with not having to apply for visa.
While Ukainian citizen might not need to apply for visa when visiting, that does not mean they have the right to work or move their residence. It just means that it will be easier to meet outside Ukraine.
This might actually boost the marriage agencies, as pro-daters could meet in Paris without trouble of visa.

However you are mixing the need of travel visa with the rights of EU citizenship here.