I don't think all forums should, it should be up to the discretion of the site owner with their niche taken into account. There's no age restriction on my forum but we do have COPPA. To date we've only had one of the forms sent into us, a 12 year old member. They ended up not really participating in the forum anyways.
Under US law you have to use COPPA or restrict under 13 year olds, but I'm from the UK so I don't use COPPA and with MyBB I cannot add an age restriction without using COPPA, so I rely on honestly and Terms of Use 😛
I think age restrictions should be applied thematically. I wouldnt be comfortable knowing that there are underage people visiting forums that use very strong and foul language. At the same time, forums about book, popular events, music...I dont see why not?
I've said this somewhere before, but honestly, even if we didn't need COPPA, we'd still put a 13 age restriction on the forum. There's less chance of this happening on smaller sites like Sakuga City, but sites like our previous one that had many sign-ups a day, we had to be vigilant of paedophiles. Anime sites and anything that's inclined to appeal to kids will attract this kind of person. We did find a few and had to report them to the authorities. There was a 36 year old grooming a 13 year old girl. She was a bit innocent for her age. That was dealt with appropriately. Anyone found to have lied on sign up about their age was banned from the site until they were 13. It might mean that they never came back, but they aren't our responsibility, especially when even at 13, sometimes a kid isn't mature enough to tell admins if someone is being creepy.
I take things like this VERY seriously and would rather not have to deal with it.
You think little kids are the worst? Try the really annoying grownups that have done more damage over time to my board. They are the worst. To be honest, I think it depends on the subject matter as to what the age restrictions should be. And besides, if a kid wants to get on a forum, they normally just lie. COPPA doesn't work.