"Years of experience that keeps us safe." vs "Years of experience that keep us safe."

I don't think you can categorically identify either version as "incorrect".

As OP says, superficially "We've got years of experience that keep us safe" seems correct, since the subject of the verb keep is the plural noun phrase years of experience.

But I personally have no problem parsing it as experience that keeps us safe. The fact that this experience is qualified by years of seems to me no different to if it had been qualified by plenty of, or a great deal of. It's the singular experience, not the plural years, that helps.

In short, both seem "defensible" to me, but I find the singular verb form flows better.