Yes. SwitchBot is the biggie but there are a ton of knockoffs (google “smart button pusher”). The catch is that the security is crap - fine for what you’re doing, but I wouldn’t use it for a garage door, for instance.

I think you’d need a shortcut or script on a computer to run a scheduled task or cron job to start it, but pretty sure that’s available too. (Note - looked at them briefly, just wanted to respond while it was fresh in my mind)

You can build your own in Home assistant using esphome with an esp32 dev board, a linear servo, and a piece of fabric that works on touch screens.

It's really not as complicated as it sounds, and if you're wanting to get into smart home stuff, HA is the way to go.

If it's on when you unplug it, does it go back on when plugged in? If so then turn off its own timer and just use a regular appliance timer.

Attatch a sausage to a soneloid and make a capacitive button pusher!

You just need a humanoid robot with the latest AI. Then you explain what you want it to do, and if it doesn't murder you or make you its slave, you might be alright.

As everyone says, what happens if you just turn on the power at whatever time?

I don't remember the brand but there are smart home button pushers for this thing. I have no idea how well they work or anything, but it's a product at least one company made at least one point in time.

I've always wondered if something like this would work:

Take a relatively short bit of wire, make a flat spiral at one end about the size of the button, tape that spiral to the button. Then take the other end of the wire hook it up to a relay with the other end attached to ground (or any big metal object probably). I would imagine then closing the relay is "touching" and opening the relay is "not touching".

I have no idea if that would actually work, but it seems to me like it should. You just need something to interrupt the electric field above the "button".

Is using a Christmas light timer not an adequate solution to this problem?

quite a few simply turn back on if you pull the power and reconnect afterwards