Try disabling any pop-up blockers in the settings of Safari, Chrome or Edge (whichever one your phone uses to connect to southwest wifi. Additionally my phone had Guardio - a pop-up blocker/unsafe website utility on it. I couldn't access the wifi on Southwest. Then on Alaska airlines a few days later I continued to have the exact same issue. I uninstalled Guardio and it felt like my phone was getting further along in the process than before. I then cleared my browsing history and everything started working.
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