Exactly what do you plan on fixing on a trail? There’s no point in bringing tools to fix something you don’t have the skills or knowledge to do. Look at the things you’d be willing to repair while covered in mud and mosquitoes, then bring the tools for that.
The stabilizer dampens the impact from potholes / rough roads on the steering linkage. That’s what it’s for. Generally firming up the steering is a secondary effect.
Phone audio + garage echo makes videos sound horrible. That noise is the throttle body performing a sweep, not the starter rattling. Sounds like you’ve got it tracked down to one problem, that’s good.