Well, there might be situations where you have don't have the choice to use your own device.
Let's suppose you work on a submarine. Or that you are a military and on the location you are the only way to access internet is in a computer that you received by your employer.
Chances are you won't have any communication with the outside world for months from that submarine. The more common scenario is an employer's computer, and indeed, I never enter my own passwords there. When convenient, I've used corporate email for personal stuff ("hey, will you buy potatoes or should I?"), but that's where it ends.
I faced similar situations a few years ago (which I don't want to specify here for privacy purposes), where i didn't have wifi in my mobile device and I had to use a computer provided by my employer to use internet. and blockchain.info was handy.
I'd use my own CD/USB, reboot into a live Linux, start Electrum, and punch in my 12 words. But I'd never enter my passwords in an environment controlled by anyone else.
Or, much more likely, I'd bring my own spare netbook and use that instead.
Checking back the previous page reveals the perfect solution:
An even better solution would be to carry around a USB stick with Tails on it, so you can boot the computer you have been provided in to a clean, live OS, and then either load your wallet from the encrypted persistent storage, or recover it from scratch using your seed phrase.