If she was tantruming over something 'wrong' on the plate, it makes perfect sense to remove the offending item briefly. Partly because it's then not obviously in front of her, reminding her of its 'wrongness', partly because she's no longer interested in it because the emotions have taken over, so will probably have forgotten it was 'wrong' when it comes back (particularly in this case because she'd moved on to focusing upon the snot on her hand being the 'wrong' thing) - and partly because if it's left within range during a tantrum, the entire plate will end up being hurled onto the floor, along with cups of water, cutlery, yoghurts and anything else within reach and then the tantrum continues because she doesn't have the food/drink anymore and has to wait for more when she's hungry NOW.
The best way in my experience to deal with uncontrollable emotions (at any age, tbh), is to not add any more fuel to the fire. Staying calm, staying softly spoken, removing anything that could be damaged, injured or hurled/smashed from the immediate area and not interrupting somebody already dealing with it.
I differ from the OP in thinking that remaining completely silent is also escalating, because purely personally, if I'm upset, the last thing I need is to feel I'm being given the silent treatment and my tendency is to do more to get some sort of reaction, at it at least means the person I'm upset with actually notices or cares that I'm unhappy, so when she started wailing 'hand, hand', I would have said gently 'do you want me to wipe your hand?' and done it myself with soft movements. I wouldn't have asked about what it was (it's bloody obvious it's snot), but asked 'is that better now?' [sniffly nods] 'are you OK now?' [sniffly nods] 'Hug or dinner first?' [Hug] 'There you go' [Hug] 'Dinner now? Here you are [returns plate, not mentioning the probably long forgotten piece of apple]. I wouldn't send a child away or force them to sit anywhere during a tantrum, as adding a feeling of rejection or of being trapped could make it worse.
When it comes down to it, as the adult who isn't having the tantrum, you are in power because you have control over your reactions. It's not being all Alpha Dog to move things out of the way. And snapping loudly as parents doesn't really help a toddler regain control of their emotions.
Any living thing, if it feels vulnerable, out of control or potentially threatened, is working on instinct, not reason. An injured cat will take a chunk out of your hand if you try and grab at it or there are other people running around and shouting at one another. An injured, scared, humiliated and distraught teenager who has picked up a knife or other weapon will be more likely to start waving it about wildly if there is shouting, raised voices, lots of people running about or somebody tries to force them to sit down or go somewhere else.
To deal with any of these things, you look for potential risks - the knife, you being too close to it, somebody between you and the exit, being bitten, a plate of food getting chucked across the dining room - calmly and quietly. Then you stay calm and ensure that the environment isn't changing rapidly around the panicking/tantruming/injured subject, so they don't overreact to it. You don't confuse or disorientate them with distractions. And you let the energy dissipate.