Loosely, you get a “use object interaction” every turn that isn’t given a lot of emphasis but is in the rules as “other activity on your turn” (pg 190, PHB 2014). It includes something like talking, opening an unlocked door during your movement, picking up something within reach from a table, or unsheathing your sword as part of your attack action. It says it should require an action only if it needs special care or presents an unusual obstacle. I’d agree that grabbing a handful of dust and putting it in your mouth could be a free action.
They used their free object interaction to pick up the tile. They’d need another action to eat it. Though going by that logic, they could just eat it at the beginning of their next turn with the same result.
It honestly just comes down to your DM style. An interaction like this is fun and has no mechanical benefit. If a player then wanted to pick up a potion and drink it as part of a free action, the DM would have to explain this to the players explicitly. But I’ve always been on the side of permissive rulings, because it allows the players to express themselves more freely. It takes more improvisation though.
But it did have a mechanical benefit!
In this instance I wouldn’t have given the intimidation, no. I’d have allowed them to put the debris in their mouth but the npc was distracted by the attack and didn’t see. To intimidate them would take an action, and a success would give them the frightened condition.
