iPhone 2G, iOS 3.1.2 and YouTube doesn't work So, YouTube application doesn't work. What I mean by that is that it won't playback any videos, nor will it log me into the YouTube so I can't see my subscriptions, or my videos/playlists, etc.

Did Google change something and Apple apparently hadn't made an agreement with them to make sure that older iDevices would continue to work with their services? Or is it something else?
It's clear now after all these years of observing how Apple handles support of YouTube and similar services, including their own Home Sharing feature, whether it's iOS or OS X, that sometimes the company is focused only on latest versions of OS or hardware. It's a sad fact of life that they don't support every single device or version of iOS/OS X but you have to give it to them because they do a very good job overall. Most of their product lineup is supported well and it just happens so that I have the first iPhone and the very first iPad that in a way were slightly experimental and very few people apparently use those devices today in 2016... so, no wonder that some features don't work at all, or well 8-6 years after the release of those devices.

Google (YouTube) is also known for changing YouTube in a way that requires applications be rewritten to remain compatible with their API. This affects not only iOS but also OS X apps, as well as programs written for Linux or basically any other OS. That's just how things are but if you run more or less recently purchased device you should be good.