I'll play along... NO! They shouldn't. Apple doesn't need to take that on. PlayStation is more open than Apple is. Whether it's a good or a bad thing, that's up for debate.
Should they already build their own gaming console? They already have... iOS products!
The playstation isn't more "open" than Apple...that doesn't even really make sense to be honest because you're comparing a product with a company. If we're talking iOS devices vs. the playstation though, the same holds true...the Playstation isn't an open product at all really. There are certain things you can do if you really get into hacking into the core build of it, but at that point, you've LONG voided the warranty, you're setting yourself up for permanent online bans from the gaming network servers (if caught), and many of the modifications can cause system faults that can leave you with a brick.
The iPhone, on the other hand, can be jailbroken without voiding the warranty...can be customized and set to do various things through jailbreak processes and applications...and if something goes wrong, 95% of the time the iPhone can be restored back to factory defaults to start all over again fresh.
So in reality, the iPhone (and other iOS devices) are a bit more "open" than the Playstation...but neither of them really fit the definition in my opinion.