Does not seem to kick in at all

Looking into the issue.

SirClickALot (Nice username btw), have you cleared your cache? that usually kicks it in, if not let me know and I'll look into it

@Doxigo → , yes, as explained in my original post. I have cleared both the Drupal caches and browser caches multiple times.

I have a temporary solution
The main issue is boostrap3 theme function is overriding the templates of sticky_local_tasks templates.

So, since you are using subtheme of bootstrap3, You can override the menu-local-tasks & menu-local-task templates of bootstrap3
with sticky_local_tasks module menu-local-tasks & menu-local-task templates. It works! (attached screenshot)

You can copy the templates from folder:
web/modules/contrib/sticky_local_tasks/templates/navigation/

I am working on fixing the overriding issue. will provide a patch if resolved.

I just see the settings button. Mouse over does not change anything.

Thanks @ Kaustab_Roy → ,

That works a treat for now!

I tried to fix the overriding issue, Found out that
Bootstrap3 has the issue of overriding the theme_registry, which is main cause of the current issue.
I have checked that by removing the bootstrap_theme_registry_alter() from .theme file, it works

But unfortunately it didn't had a proper solution. So adding this for reference.
https://www.drupal.org/project/bootstrap/issues/2284283 →

Thanks everyone involved, I consider this as a works as designed situation, closing it