Possible use of mid-2011 iMac for display with 2021 MacBook Pro?

Not via Target Display Mode. See:

Use your iMac as a display with target display mode - Apple Support

You iMac is capable of TDM but your Macbook Pro and its OS are too new.

I'm seeing reports that Apple's Screen Sharing and third-party solution give unacceptable lagginess. A standard external monitor:

• is cost effective
• offers more options
• won't lag
• is more power-efficient
• is easier to set up

Also the cables and adapters required to make an old iMac work with a new Mac can cost as much as a decent stand-alone monitor.

Hi, rahneo, although it’s technically possible( via DuetDisplay/LunaDisplay) it’d be cheaper and more practical to just buy an external monitor or connect your 2021 MacBook Pro to a flat screen tv set to use as a display

John B