I tried lots of things but was eventually checked the Brother website. Here is the path I took:

1. https://support.brother.com/g/b/countrytop.aspx?c=us&lang=en
2. Click "Downloads"
3. Enter MFC-L2740DW
4. Select macOS 12.x
5. Select scanner driver (ICA) 10/28/2021(3.5.1)
6. Download and install

Reset the printing systems again, and add the printer again. This time when it was installing there was a scan option.

Dragged new printer object to dock.

Scanning seems to be working again.

I tried lots of things but eventually checked the Brother website before seeing the first answer. Here is the path I took:

1. https://support.brother.com/g/b/countrytop.aspx?c=us&lang=en
2. Click "Downloads"
3. Enter MFC-L2740DW
4. Select macOS 12.x
5. Select scanner driver (ICA) 10/28/2021(3.5.1)
6. Download and install

Reset the printing systems again, and add the printer again. This time when it was installing there was a scan option.

Dragged new printer object to dock.

Scanning is once again available. As noted in the edit to the question, I also needed to again reboot the Mac to get it completely working.