OK, I’ve been waiting to answer this until I had enough energy to keep this from being a hugely long post. (Sometimes my most tiring struggles are with myself! LOL!)
capacitance: the absence of charge
Most people think a touchscreen is somehow measuring electricity brought TO the screen. But it isn’t. Instead, a touchscreen responds to interference with the small amounts of current that were already there.
In order to get that block, you have to have something which can conduct electricity AWAY from the screen. And it has to carry enough away that the touchscreen will recognize it as interference.
This is why pretty much anything that is conductive held by a human hand will work, because the electricity gets carried away from the touchscreen and the human body is big enough to absorb enough of it to register as interference.
But just laying a metal coin on the screen, even though that coin is conductive, won’t register because it doesn’t block enough of the current on the screen.
It’s also why a long wire will work better than a short wire, especially if it gets grounded in some way.
OK, that out-of-the-way, now we know what we need: something that can touch the capacitive button and carry away some of the existing electricity.
Real World Applications
There are a lot of these used in manufacturing touchscreen devices: that’s how you test them at the factory. But those are usually $100,000 machines and pretty big. Not practical for home automation. 
There’s a small market for people with varying degrees of paralysis, but to be honest, these days most of those devices, including tablets and smart phones, come with other methods for controlling them, like voice. So this market is smaller than you think.
There’s a fairly significant demand for this among robotics hobbyists, where they are often called “monkeybots,” although that term is also used for other things. But the idea is to have a robot that can use a capacitive button. I’m going to talk more about these in a section a little further down. They are almost all individually built, but you can find a lot of plans for them if you look for “robot that can use a touch panel“ or a similar search.
Then there’s home automation to retrofit exist devices, like your air purifier. The thing is, most home devices that have a touchscreen, as @GSzabados mentioned, also have a remote, and once you have a remote, life is easy. If nothing else, you can just use a Switchbot to push the remote buttons.
Your device is unusual in that it has a capacitive button but no remote control capability. This kind of device was popular in Asia for a few years, which I suspect is why Naran added capacitive touch to microbot Push. But most newer devices offer either a button remote or voice control, which I suspect is why Switchbot didn’t bother, because they were more about “good enough“ engineering at a budget price.
A monkeybot example
Here’s an excellent example of a robot built to use a capacitive tablet. Note that most of the mechanism is there to move the stylus to the correct position on the tablet, so we don’t care about that. Our project would be fixed position.
Also note that while there are multiple batteries and wires and all that kind of stuff none of that has to do with actually activating the tablet. That’s all there to move the stylus around physically.
What’s being detected by the capacitive screen is a fairly long stylus with an even longer piece of wire attached to it which is then grounded. They put a nail on the bottom of it, but they didn’t need to. They just needed a conductive tip. They could’ve used aluminum foil, copper foil, or just bought a tablet stylus that already had conductive rubber on the tip.
Modifying a switchbot
OK, now that we know what a capacitive Sensor is detecting, it’s pretty easy to imagine converting a Switchbot to work.
You need a conductive tip, and I would just use copper foil because it looks nice and conducts really well, but you can also use aluminum foil.
You need a long wire, probably 12 to 14 inches, to go inside the foil making contact with it and then running out to the ground. A metal shelf brace would probably be perfect, you want something maybe 6 in.² or around that in order to get enough of a grounding effect. Or you could just ground it to a small metal trashcan. It’s not going to carry enough charge to hurt anybody. I would not ground it to the air purifier itself.
Now when you actuate the Switchbot, the foil touches the screen, the electricity already flowing through the button gets carried away up the wire to the ground, and that should be enough of a block that the button registers it as a “touch.“
The thing is you want to make sure that none of the foil is touching the button except when the Switchbot is extended, so you’re going to have a little bit of trial and error with all of that.
I suspect this is the main reason that most people who have project reports on this have the Switchbot pushing a stylus which then makes contact. It’s just because it’s really hard to get the foil over the tip of the Switchbot itself but keep it isolated when you don’t want to be touching the button.
So, yeah, if it was me, I would probably buy an inexpensive tablet stylus, wrap a wire around the tip and run it up to a ground plate, mount the stylus vertically in a little spring holder, and put a Switchbot at the top to press the stylus down far enough to make contact with the button on the air purifier. But that’s me. It would look really weird, if there are kids in the house they would probably want to play with it, and it’s not the direction a lot of maker type people would take.
My guess is people with stronger aesthetics than me would take a Switchbot, mount it on a little platform, build up the tip, and do it that way. It would look a lot nicer than mine.
But the main point is all you need is a conductive tip, a wire to carry away the electricity for far enough distance, something to attach the wire to to ground it, and a way for the Switchbot to make the tip touch the capacitive button. Switchbot operates on its regular battery in its regular way, and you don’t need any additional power for the project. Your goal is to carry electricity away from the touch panel, not to it.
@ogiewon or @johnconstantelo might have some ideas.
