Hi,
There is a major script error on the error console: https://i.imgur.com/EhtstBU.png
That blocks further script execution and the plugin can not initialize after that.
If the search works when you are logged in, then this is an issue with your cache for sure. If you are using some sort of a script minify/defer feature, that can be the cause of that error message. On the reported error srouce I see an inline variable, which tries to access a property, but the script where it is initialized is defered, so it does not exist yet: https://i.imgur.com/bk2rkKE.png
Make sure to always monitor the error console when using script defer or async features – those need to be setup very carefully.
It works after the redirection because on that page the error message is not generated as the defer attribute is not there, I assume because of the query arguments, so the plugin initializes correctly.