Need ideas for simple closed circuit tv for church overflow room
If you want to remotely control the pan, tilt, and zoom of the camera, you want to look for a "PTZ" camera. You will not find one under $500 with a controller.

A cheap camcorder with a wall mount and a long HDMI cable is your cheapest option.

If your budget was closer to $2,000 you'd be able to get everything on your list.