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I'm wondering if a switch exists that fits this functionality I imagine:

The user would depress the switch, it would latch and stay depressed. Once the product completes its function, the control circuitry sends a signal to the switch which un-latches it. I imagine the actuator could be spring loaded so it would pop back up, or maybe a solenoid pushes it into the open state.

I wouldn't even mind having to supply current to a solenoid to keep the switch depressed. That is what most toaster actuators do. In fact, what I want is exactly what a toaster does. But I want to avoid investing time in designing a custom mechanical solution for this.

Can anyone confirm that such a product exists in a form factor like a standard panel mount switch?

Some words I can think of to describe this are:
Electrically toggleable switch
Latching switch with electric reset
Switch and latching relay combination
Electrically controlled switch

I've searched around with these and other terms, but haven't found anything.

Here is what I imagine the circuit symbol would look like:

Thanks for your help.

Figure 1. A solenoid hold-on pushbutton switch by EMC.

There are solutions available but they're likely to be expensive and hard to come by.

Figure 2. Momentary illuminated PBs. Random web image.

Once there's a micro-controller most of us would just use an illuminated momentary pushbutton. Illuminate the LED using a GPIO output when the button is pressed and turn it off when the time is up. The whole world has been conditioned to accept this kind of switch by now.

The only reason I can think of for the solenoid latching switch is that you might need it to maintain its state during a power cycle.

Can anyone confirm that such a product exists in a form factor like a
standard panel mount switch?

What you are describing is most likely not commercially available. The reason for this being is electronic switch are easier to make these days. Search timer pushbutton switch. Most of these will not have the mechanical functionality you describe. Another thing to consider would be a rotary timer switch which allows you to set the time.

Older toasters use a bimetallic strip that is temperature activated to actuate the toaster. I'd imagine that newer ones use solenoids.

If you really want to get a switch with functionality like this, your best bet is to build one yourself with a commercially available solenoid for actuation.