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Beachballs can be many things including slow/dying HD or possibly too many daemons running in the background. My 2011 27 in. iMac still runs just fine. My steps would be:

  1. Download Etrecheck to find out if you have unexpected/unwanted background processes
  2. Run Apple's hardware test to see if there is any issue. If you have other tools ( TechTool Pro and Smart Utility come to mind ) use them to evaluate the HD.
  3. Install an external SSD. You can connect it via a sled and a Thunderbolt cable. It will boot and this will avoid opening your iMac. This is what I use but there is probably something newer and less expensive( newer version here )

Sorry, but if that's the cost for adding a Thunderbolt SSD, it's not at all worth it, since internal SSD drives are so cheap these days. I've done SSD surgery on this model iMac; it is not hard at all.

The other problem I see is that you're leaving a (probably) faulty hard drive in the machine. Even if you stop using the drive by moving all the data off of it, it's still connected to the system and bogging it down, no? And it's certainly adding plenty of heat, which makes the fan run, which is noisy. I love that my 2010 machine runs fast, cool, and quiet because of the SSD in it.
 
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Sorry, but if that's the cost for adding a Thunderbolt SSD, it's not at all worth it, since internal SSD drives are so cheap these days. I've done SSD surgery on this model iMac; it is not hard at all.

The other problem I see is that you're leaving a (probably) faulty hard drive in the machine. Even if you stop using the drive by moving all the data off of it, it's still connected to the system and bogging it down, no? And it's certainly adding plenty of heat, which makes the fan run, which is noisy. I love that my 2010 machine runs fast, cool, and quiet because of the SSD in it.
The hdd fan runs anyway, and that's a good thing. I even left my odd fan running, but disconnected the odd. Given the iMac desing (intake from the bottom and exhaust from the top) even if it's not direct air over the interested parts, it still helps moving the hot air outside of the tight and close space.

But yeah, you have to swap the hdd (which makes most noise). Thunderbolt ssd is not the perfect solution.
 
I was thinking more in terms of something like this I found on Amazon. Can I go straight to El Capitan from Lion?
 
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If your mac supports 10.11 yes you can got from 10.7 up.

I have done this on my 2007 imac.

but if you have internet and app store do it via app store.
then create your own usb media as it will be free!
 
if yours is 2011 you don't need to upgrade, let's be real here. it has a good screen, it has a i5 processor (core 2 duo is obsolete, that i5 will be pretty modern for a while). just double the ram, put in the ssd (which isn't hard at all) and you got yourself a little beast (which will beat almost any current iMac that still has the spinning hdd). i kid you not, it will be twice as fast if not faster.
I agree. 2011 is the last year before the "super slim" iMac design and that/your iMac model came with Thunderbolt. It's not that old. The Intel processor is modern. You have a few more years left, for sure. Grab an SSD, upgrade the RAM and update that OS and it will truly feel like a new machine.
 
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