This always seems to happen. You’ve got a NPC refusing to offer some assistance or key information. Then a player grabs a d20 and tells you they are cuffing the guy across the face and demanding something, rolling off using their intimidation skill. It’s at a point like this where I might turn the NPC into a gibbering heap, spouting snot, tears, and nonsense, completely worthless for offering any help. That demonstration of physical force and the threat of more has rendered the person panicked in utter fear, unable to act. Or alternately, I might even push an NPC into responding with violence as they feel they have no doubt the player means them harm.
I always viewed intimidation as more than just physical threats against someone. I see intimidation almost like insight, but keyed in on sensing weakness. They are able to read that weakness and then use it as a tool to force someone to their whims.
It can certainly be a physical aspect but it doesn’t have to just be a warrior hefting a weapon threateningly. It could be a stare or a physical presence that instills doubt. We’ve likely all experienced this before. You’ve met someone that just carries themselves in a way that projects threat.
Take the crime drama, Knockaround Guys. There is a scene where Vin Diesel gives a beat down on a local thug in a bar. His dialog at the start of that scene is great. It’s just him talking about wanting to be a tough guy and figuring as a kid 500 fights was the number needed. He describes his reasoning and simply starts taking off his jacket. You can see the steam and ego of his opponent just deflate. The local thug has overstepped the line. In front of him is a legitimate tough and all of this is achieved through words and an imposing physical presence.
Intimidation is the key ability to know what threats will work and what will sting the most. It’s an ability to read the social standing of someone, and likely what they hold near and dear. Threats don’t have to be physical. They can target a NPC’s character and their standing with the law, or maybe a threat to dishonor them in society or to family (think Francis Urquhart/Underwood from House of Cards).
Someone with a high skill in intimidation can also likely read when a threat is serious or not. It’s staring right through a facade of a violent display and recognizing it for being just chest thumping and bluster. So PCs with this skill could read certain threats and certain social situations much like insight would.
Unlike diplomacy, that seeks a common ground and tries to build cooperation through good will, Intimidation is all about manipulating people through threats (real or perceived). Not all of these threats have to be physical. When you look at intimidation as a means of seeing weakness in someone, and being able to capitalize on that, you open the door for a lot more opportunities and a variety of means to use the skill. Consider letting your PCs explore intimidation being more than just the ability to get answers from someone by knocking out teeth.