Researches of the whole site would be better instead of focusing on its customer support. Customer support does not guarantee that the site is a good place to spend your money. At least try to find some valid reviews about the site where you are going to play.
I do agree with that we need to choose a gambling site which has a good customer support but before that, we need to check whether a site is trusted or not. Among few trusted sites if you want to choose a site to gamble then you should choose the site which has a good support team. There are chances that we may face some issue if we plan to gamble regularly so if support is not responding faster then it will be an issue later on.