I assume you want to play games online. Have you considered a trial run with something like a game console or PC without network access? (Is it the Wii Slim that doesn't offer Wifi?) Many parents concerns these days are about exposure to the internet and a fear of 'always on' dependencies. I'm watching my brother go through it with his toddler already and that smartphone sure is a handy tool when something needs to get done...
My only real argument is this. I would rather have my child learn personal responsibility now than later. I know my early college days suffered from too much freedom and a re-evaluation of priorities had to take place. Better to do that when schooling is correctable and with parental oversight than wasting tuition and potentially housing costs. I also note that a lot of social engagement with peers is done online in a contemporary setting. Being the one kid that doesn't play the hot game can actually have negative impacts.
Only other potential argument is if you have any plans on making anything to do with computers your career. Then an argument could be made for using it for more than gaming. (Though really you don't need a gaming PC to do that). But I have run into people that say they learned a lot with Minecraft and others who learned almost all their programming skills from writing game mods.
I got my first second hand PC (486) when I was about 10. Of course this was the mid-90s and the internet really hadn't hit the Midwest yet. I used it almost exclusively to play games, but when it couldn't handle what I wanted, I started upgrading it. OS installation, adding expansion cards, etc. Now back then that was significantly harder, and not having easy access to the internet meant a lot of trial and error. While my career doesn't have much do with computer hardware the logical approach I learned in having a computer to troubleshoot is invaluable to other applications. I now have a career in software asset management for a fortune 500 company. (Started out with upgrade projects from Windows XP to Windows 7, quite the feat for companies with some software being over a decade old)