I have an iMac from mid-2011. Why can't I upgrade to a newer system? Also, the required upgrade to import photos from my iPhone does not download. Why?

It cannot beyond High Sierra because newer versions of Mac OS require Metal, a function built into the

GPU. Metal became available in iMac GPUs in 2012. Before you ask, no your GPU cannot be upgraded and no there are not any work arounds. If you want or need Mac OS beyond 10.13.6, it is time to replace your current machine.

https://everymac.com/

lists specs for old iMac models and indicates what version of the macOS originally shipped and which is the latest version each one can support. Those from 2011 or earlier don't support the current version of macOS.

You might be able to sync your photos from your iPhone to your iMac using iCloud, or just email yourself the photos you want to get from your phone onto your computer, if you just have a few of them.

That Mac is not compatible with anything higher than High Sierra.

What required upgrade? iOS 13 ought to be compatible with all Macs running El Capitan and newer. I also just helped my fiancée import her iPhone photos to a 2011 iMac running Sierra.

Is the iPhone connected to the Mac via a USB to Lightning cable?

https://support.apple.com/en-us/HT201302#importmac