Hi,
Thanks for the details. I tried to test, reloaded the page a couple of times, but all worked okay on my end. From the video, it looks like as if the search script was either not loaded, or interrupted by an error. I fired up the javascript console, and there are indeed errors reported from 3rd party script files: https://i.imgur.com/CWWdVSO.png
I strongly recommend resolving these, it is very likely that the issues interrupt the script loading, depending on the client timing.
If the issue was related to the plugin, it should be possible to replicate 100% of the time, as the script does not change. I am also noticing the cloudflare rocket loader activated, that can also mess up the loading times if there are script errors.
Once you resolved the reported issues, make sure to clear all layers of cache, page, CDN, assets, browser. That should resolve the issue.