Thank you, I indeed missunderstood you.
I found the issue, sort of a bug, but it is more complicated. So basically, there is some sort of a script injecting some the content to the page (including the search), and the injection was not yet completed, when the scripts were loaded. The script was trying to initialize any search bars on the page, however during the execution none were present.
The jQuery version worked almost all of the time, because that is much slower to render for the browser. On some refreshes even that did not work, when the timing was right.
Luckily it is not a big deal, as we encountered similar behaviours with the legacy scripts a few years back, and we added this option to resolve these types of injection/late loading conflicts. It is extremely rare, but yet it can happen.
I have turned that option on, and it should be fine now with both script sources in every case.