Google has declared EOL for the original Pixel. I got my Pixel for a Google Fi account I use just for travelling, and I'm not really interested in replacing it. Realistically, how much longer can I use my original-release Pixel before it stops functioning?
Theoretically, the OG Pixel will work indefinitely until there's an irreparable software/hardware issue. Realistically? Depends on how the user takes care of it ;)
"End of Life (EOL)" doesn't mean that the device will end its life. It is just that the manufacturer (in this case, Google) will stop its support; no more security & OS updates, and the like.
If the user is adventurous enough, they can unlock the bootloader (warning: will delete all the storage) and flash custom ROM which at least extends the life a little further. (In addition, seems it's also compatible with Project Fi).
Anecdotal evidence: I'm using Nexus 5 which was released in 2013 (Android 4.0 Ice Cream Sandwich) and EOL in 2016 (Android 6.0 Marshmallow). I flashed Lineage OS after that until it's not maintained (Android 7.1.1 Nougat). As of 2019, the phone is still usable in daily life.
If Google declares a device to be "EOL" this simply means that you no longer receive device and security updates.
Therefore your Android devices will not stop working, however after the point of EOL the chance of a security vulnerability that allows to attack you and your data will continuously raise.
However as Google devices can be unlocked easily you don't have to accept this. Unlocking the bootloader allows you to install an OS that does not come from Google, like LineageOS (warning: unlocking will erase all user data). The first Pixel (code name "sailfish") is a supported device by LineageOS. Therefore you can download and install LineageOS on your device and again receive updates until LineageOS also drops support for your device.
But there are also some draw backs of unlocking and installing LinegeOS: Some device assume that LineageOS devices are rooted and therefore insecure (they just assume that in general, independendently of the reality). Therefore apps like GooglePay may refuse to work.
In reality, i'd say a few years until it loses his usefulness. I have a 2 old phones that still work if plugged in, but a lot of the apps now fail because the API endpoints changed. Google maps won't load any maps or autocomplete searches, and only shows me a dot on a grey screen if GPS is turned on. YouTube videos wont play anymore, neither with the official app nor via browser. I think the webbrowser has some issues with SSL certificates that are now required everywhere. In the best case you get a message to update your software, but then the google PlayStore tells you that the latest app version is not supported by your android version.
The hardware, phone and text messages might still be working in 10 years, but eventually your apps wont be supported anymore.
In one respect it became a potential brick the day after your warranty ended, as it'd doubtless be uneconomical to repair. But it's not going to get 'turned off'. It'll work until it breaks.
I have a Pixel C with unresponsive screen. Effectively a brick. Pity. It was a nice machine.