I'm going to try and cover every single reason OP and I'll let the more technical people elaborate on them.
First though, it wasn't AMD Intel NVidia. Since we're going to go back in time and talk about something like Larrabee you have to realize it was AMD vs Intel for CPUs and Nvidia vs ATI for CPUs.
In 2006, AMD acquires ATI with a new idea for the APU! Great integrated graphics performance.
Before this, AMD had just released it's most competitive architecture yet that swept the floor with the Netburst Pentium 4s.
In 2006 AMD acquires ATI and focuses on graphics performance.
At the same time, Intel focuses on CPU performance and releases Conroe.
Subsequently, we got one of the most revolutionary processors that destroyed AMD as AMD's focus was on Graphics (HUGE MISTAKE). Since then AMD has never recovered or caught up.
Now ask yourself, why would intel, seeing how in the past, a CPU maker who focuses on graphics dug it's own grave, why would they follow that same path?
Now lets move on to actual performance. Does performance matter when selling a product? In some aspects yes it does but mostly it doesn't. Just ask intel again, they sold the Pentium 4 and had I believe to be 75% of the market at the time with a HORRENDOUS architecture. AMD had around 25% I believe with a far better performing processor. Did it matter? No.
Now lets fast forward to today and look at AMD vs Nvidia.
To get some quick dirty stats here (not representative of the whole just of gamers)
http://store.steampowered.com/hwsurvey/videocard/
AMD has 30%, Nvidia 50% and Intel 20%. Yes, 20% among GAMERS (how can you even game on an igpu?). So despite AMD having a VASTLY superior product to Intel for GPUs, they are only 10% ahead.
But the real question is why is Nvidia at 50% of the market? Simple, name brand recognition. Plain and simple, AMD has released better GPUs than Nvidia multiple times and yet Nvidia almost ALWAYS outsells them. Not even that they outsell them, it's that Nvidia can charge more money for the same performance than AMD. The only way for AMD to compete with Nvidia is to compete on price to performance. Consumers don't purchase based off graphs on a hardware site (Very few anyway), they purchase based off name brand recognition.
Now lets jump to Intel's move into the mobile market, specifically tablets. AtenRa loves to laugh at Intel's Contra Revenue scheme but it's the ONLY way intel can get into the market. Even though intel does have a competititve processor for tablets, they almost have to give it away for free to be able to be used. Why? Because even when the consumer is a business, they don't want an unknown. They want something that is known and will pick a Snapdragon Processor 95/100 over Intel's Baytrail, Clovertrail, etc. because they know it.
Conclusion:
Intel focusing on GPU would be a massive mistake.
Cost: It would take a TON of cash to get the facilities, research, and marketing to bring a GPU to market for intel.
Benefit: The Revenue AMD makes from GPU's is a drop in the water compared to the ~$15 Billion in revenue per QUARTER intel makes on CPUs, hence why they focus their efforts their (Nvidia who sells more GPUs than AMD hits around $1 billion a quarter in total revenue for reference).
Performance: Even if you have the best performing product, brand loyalty matters. Just look at the number of users in the VC forum who upgrade YEARLY to the new SLI cards. Nvidia sold a single card for $1000 no issues... have AMD or even Intel try that.
Etc....
In short, not worth.....
Edit: 2013, Intel pulled in 52 billion in revenue. AMD pulled in 5 billion in Revenue. Focus all their money for a market that isn't even 10% of their revenue? No thanks.....