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:
- Download Etrecheck to find out if you have unexpected/unwanted background processes
- 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.
- 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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