Can I use a switch to perform momentary actions and also cut its own power source?

this guy can turn him self off:
and this one is similar to multivibrator:

i see no reason why proper control circuitry couldnt do the job even if device is switching its own power, i have seen delayed trigger circuits in walkie talkie buttons that compose of resistors capacitors and transistor that "could" work for you, personally as always I think its easiest to use microcontroller and code,

if it were my choice i'd select this type:
3) A double-coil latching type where one coil latches it and the other coil unlatches it,

however it would be trickier for device to turn itself on if it has no power to run control circuits, and , you need to only pulse the on coil briefly , then shut it off so it unlocks so it can be ready to be triggered by the off coil,

**broken link removed**
if you look at a flasher circuit(or multivibrator) it is similar to what you are thinking but idk if you would want to do that your way without putting a delay on the "switch on" coil as well, other wise you will get short off pulses and longer on pulses.... as for doing it with relay instead of transistor .... maybe...
I do know it will shorten life of relay to be pulsing it all the time, relays are not meant for "switching all the time"

Here's how you could use a dual-coil latching relay to provide power to a switch which, when operated, will remove power to itself. Is that the sort of thing you mean?

A latching relay is a bistable operation which latches in each direction.

Agreed (if by 'in each direction' you mean remain set and remain reset, until the appropriate coil is pulsed).
So why do you think the post #22 circuit won't work to provide power to the NO switch from the time the latch coil is pulsed briefly until the time the NO switch is closed momentarily to energise the unlatch coil? I'm assuming (perhaps wrongly) that when the unlatch coil overcomes the latch permanent magnet and begins to flip the relay armature, then armature inertia plus the coil pull will continue the unlatching action until the relay contacts supplying the NO switch open, thereafter the armature inertia plus the unlatch permanent magnet pull will take over to complete the unlatching. Is that assumption incorrect? I admit to never having tested it in practice.

Alec No to your #22

Panasonic indicated it is bad (X) practice .. see fig 22

https://media.digikey.com/pdf/Other Related Documents/Panasonic Other Doc/Small Signal Relay Techincal Info.pdf

Hence false assumption that self-interrupting contact to coil current on any latching relay will continue to toggle.

Other designs may have exceptions, but remember this must Break before Make to avoid shorting out supply from transient arc and bistable may have strong force for fast switching.

Thanks for the link, Tony. Something new learned .