Is there no good full screen Remote Desktop App?

Have you tried the built-in Screen Sharing?

To Share [read: Make available for others] use:

System Preferences > Sharing > Screen Sharing ... to specify who can connect.

To start to use the Shared screen, The Application here:

/System/Library/CoreServices/Screen Sharing.app

can be dragged into the Dock for easy access.

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I do not understand how clamshell mode can possibly figure in the problem you described.

It would help Readers if you broke your thoughts into paragraphs instead of the dense stream-of-consciousness you presented in your original writing.

If you want to use an external display, the best way to do that in a small space requires blocks of wood. Elevate the external display so that the MacBook can remain open directly below, and you can use the built-in keyboard and trackpad, as well as the additional space on the second display.

I am not a fan of closed-clamshell mode, and the cooling is not good with the lid closed.

The default for Macs is to add the External display to an edge of the built-in display and create an Extended Desktop. This picture gives an idea of how it works. The tiny screens can be dragged to represent the actual positions of the two screens on your bench, and the tiny MenuBar can be dragged to either display. The mouse moves seamlessly across the boundary between the displays, and can drag a window with it.

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