The problem is caused by the Gemini web application automatically redirecting the user from its root URL (https://gemini.google.com) to a specific application path (https://gemini.google.com/app or https://gemini.google.com/u/<id>/app).
This internal redirection mechanism interferes with Violentmonkey's script injection timing when a new tab is rapidly opened by the «NewTab Redirect!» extension.
Consequently, the userscript fails to execute on both the initial and the final destination pages, regardless of implemented delays or intermediate redirect pages.
I have solved the problem by updating the «Redirect URL» option's value in «NewTab Redirect!» to point directly to the final application URL (https://gemini.google.com/app or https://gemini.google.com/u/<id>/app) instead of https://gemini.google.com.