Let's say I used the thaumaturgy cantrip to create a booming voice, make the ground tremor, and start a rumble of thunder while suggesting someone to do something in an attempt to intimidate them. Would that impose an advantage on the Intimidation check? Maybe not so much rolling twice, but a bonus of sorts?
This is also assuming they are not creatures that understand magic to the point where they know what's happening.
One of the advantages (haha) of Advantage is that it can be granted circumstantially, as per the rules on Advantage:
The GM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result.
So if the DM says that using Thaumaturgy grants Advantage on intimidation, it does.
And if they say it doesn't, then it doesn't - this isn't even a matter of optional rules, it is entirely within their power to decide if advantage should be granted or not.
By RAW, not particularly.
But in RP? yes, those are generally seen as intimidating effects, and its up to you/the DM to negotiate how that will be implemented. It could be by directly giving advantage or a positive bonus to your rolls, or maybe by lowering the DC or giving the NPC disadvatage.
The rules on giving circumstantial advantage are a great way to reward cleverness, innovation, and teamwork by the players as they try to set up situations where advantage would accrue. I had advantage awarded to my cleric for doing something a lot like that with Thaumaturgy, in our first campaign. Learning from that, I've looked for innovative uses of various skills to do likewise.
The DM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result. (Basic Rules, p. 60)
Using Thaumaturgy that way, if the player(s) set(s) it up, would be one way to fold that higher level concept - applying circumstantial advantage / disadvantage - into mechanical results in play. The key is in listening to the players as they respond to the basic flow of the game.
The DM describes the environment.
The players describe what they want to do.
The DM narrates the results of the adventurers’ actions.
Setting a climate at the table where coming up with clever solutions to difficult situations is encouraged, and sometimes mechanically rewarded, is a place where this edition shines. But it takes more than looking at an If/Then rule statement: context and set up, like a good joke and its punch line, are a key to getting the most out of each opportunity.
