Except...the USA is not the leader of the free world. Sure, that's what they like to call themselves, but exactly how much does the free world really listen to them?
Let's look at the developed world:
Canada - sees the US as the crazy cousin nobody invites to family gatherings
Every country in South America - sees them as nothing more than a bully
UK - usually follows along with foreign policy, rolls their eyes at US social policy
Europe - laughs at them heartily, about pretty much everything, but especially human rights and food standards
Far East - the US are useful customers to keep their economies ticking over, nothing more
Asia - the US is a target whose lunch needs to be eaten, just like the UK's
Australia - keep your crazy, we've got enough here already
NZ/Pacific Islands - LOL, you call that sport?
Hell, the most chest-beating, USA-is-star-spangled-awesome political party's main talking point is how bad the US is at democracy (regarding the loss to the Democrats last time round) and justice (how Trump's getting his ass handed to him in the courts). And we all know that quality of life in the US - access to healthcare, employees' rights etc - is lower than just about everywhere else in the developed world.
So no, they're not the leader of the free world, but it's handy for the population to believe that while their politicians and industry are stripping them of all the rights they'd have almost everywhere else, in return for hanging on to their guns.
When considering their influence on the world, you also have to consider where they actually sit on the political spectrum. Even our current insane-right government in the UK would be considered moderate centrists in the US. There is also the fact that their judiciary is a political body, not a purely legal one, and stacking that particular deck is how they're getting this stupidity into law. That can never happen here, because our judiciary is fully independent of our political system.