Chrome is a 3rd party application. They have their own support cylcle and it is unfortunate it does not match Ubuntu's support cycle. Canonical is not responsible for this browser.
Do Google not realise 12.04 had a 64-bit version? Or will updates eventually come to an end on this machine?
Yes, they do. The notice they made is not just about 32-bit but also about Ubuntu 12.04:
- Google Chrome is to drop support for all 32-bit Linux distros from March, 2016.
- Google Chrome will also drop support for Ubuntu 12.04 LTS (which will receive critical and security bug fixes from Canonical until mid 2017).
It is their choice to kill of the 32-bit version. Probably saved them money and resources so it might even be a logical step.
So if you want to use Chrome and use a version that has all the upgrades your options are to upgrade to 14.04, 15.10 or the upcomming 16.04.
By the way: the browser for Ubuntu is Chromium. Canonical takes care of this browser during the support life cycle of Ubuntu.