Connecting a 2011 macMini to a new iMac

You could boot it into target disk mode and connect it via Apple's Thunderbolt 3 - 2 adaptor with a Thunderbolt cable to access its hard drive (it will show on the iMac's desktop as you would like), but again, that would not be a good time machine destination (not even sure if macOS would allow you to if the mini still had its OS installed, but it probably would if you erased it).

That said, 2011/12 minis are still coveted machines due to upgrade paths, you could be better suited selling it, putting some proceeds towards a dedicated backup drive, and pocketing the rest! 🙂

It seems like you know your plan, but macOS Server - Overview - Apple (CA) is a nice little product with a host of features, that may allow you to get more use from the machine. I use it for its time machine and vpn services.