My ps2 has Ethernet, so does my Xbox 360. They're two different phases of online access. The ps2 I wouldn't worry about putting on the Internet. The 360 is a bit different because it has firmware updates. That said, afaik Microsoft has not tried to remotely disable the 360, which is about 20 years old. Xbox live even worked a couple years ago when I really wanted to play rez. I think they disabled xbl finally but they didn't brick my console or make it not work less.
So it's a choice.