Running macOS High Sierra Without GPU

Well the computer will not run without a GPU so if that is the goal you have in mind it simply will not work. Considering the 2011's have be classified as obsolete by Apple for several years now getting parts may be a challenge. In addition, 2011's will not run current versions of Mac OS. At the end of the day, the computer is model 11 years old now and has run its course. You can put money into it, however I think you would be wiser to put that money into replacing it. Even if you fix it, it would be operating on borrowed time, it could last a another 5 years or completely die within days of repair. This is why I'd not put any effort into it.

I also have a 2011 27" iMac and fortunately mine is working fine however as I know it's close to end of life I intend to replace it within the next several months. If finance is a consideration , then consider Apple 18 month free financing and you can also search for an Apple Refurbished iMac on Apple's US site. The refurbs have zero down side, they are identical to new, come with the same 1 year warranty as a new Mac, are eligible for AppleCare + and can save hundreds of $.

Best of luck.

...could I remove the GPU and run off the Intel integrated graphics?

iMacs do not have dual graphics. That is a feature of MacBook Pros.