I had this on my old forum years ago, but got rid of it months later as it served no real purpose (even when I had post count disabled for it). Now I don't include it at all.
I think the games forums have a place within General Forums and perhaps gaming specific niche sites but outwith that, they don’t serve a lot of purpose on other than bumping post count and producing low quality posts.
On my last forum I started without forum games but ended up adding it when members asked for it. They can sometimes be fun. But I made rules against 10 different counting variations etc.
I've had them on my forum since day 1, and people like to post there. It's a good way to get people chit-chatting without a deep commitment to replying to longer threads, and often after a bit of games, they start posting elsewhere on the forum.
That's what I did when I first joined forums too, so it makes sense for me to keep them around. Plus, I enjoy the games too!
I deleted my forum games forum due to receiving little to no activity in many years. I own a resource board, so most people go there for the resources.
If you go to any forum that has posting games, you'll typically see daily contributions by members of the forum on those types of posts. Without those posts, your daily activity would likely be a lot smaller. Those types of posts help community members get to know each other, have fun with each other, and become loyal to your community.
Not everyone wants to talk serious all the time.
Posting games gives people something to do when they don't want to be serious and it still keeps your forum active in the process.
I say create an entire section just for postings games.
I do have a few selective bored games, as I like to call them. Some of them can be interesting, but I only really like the more creative games, but that fits in with my niche of Creative Writing. Games like Count to 1000 serve no purpose other than to boost post count, and that's great, if your site uses post counts as a means to access higher functionality. Games like Corrupt a Wish, games that cause a person to put some thought into their posts may add value. I've hosted games like the Alphabet game, but I usually put a stipulation that we have to use Anime Girls name or something. Six Word Story can also turn into a fun madlib adventure.
Technically, this could fall under Gamification. But I've found that using alternative methods to gamify my hobby work better than posting a bunch of what may be considered spam.
We got rid of it at CH early on because one guy joined and did a counting thread by himself so he could have a larger number of posts than everyone else. He got mad when we removed it and never came back.
I never really had an issue with them on forums. They were a nice way to keep activity up, and it was a nice little way to get some posts in if you were wanting to be active without having to get into deep conversation and just wanted to have a quick little forum visit that day. Unfortunately, it does kinda look bad when that's the only place people are really posting and not really spreading the love elsewhere.
I've seen it where posts do and don't count towards the post count of users on the site. Doesn't really bother me either way if it does or doesn't (I think I moved past the phase of life where I cared about my post count lol).
I grew up with niche forums with forum game areas so I have never minded them. I get the whole about feeling like a waste of time for some admins so I don't mind if other forum owners don't include them. I think they are easy enough to hide or avoid if it's not your jam.
I like to have the zone in my community because they helped me a lot back when I was new to the forums and not completely fluent in English. The forum games were a great ice breaker and a way for me to get to know some of the friendliest people.
Heck, I met my partner on the forum games area of a Zelda forum we both joined the same day.
Now I a married and have been living in a an English speaking country on the other side of the world for over a decade.
That's the bottom line for me. In recent times forums tend to be run more as a service and in that respect posting games are not really appropriate. They are better suited to general chat style forums.