First handle on first anchor point disappears

Tommy,

"The only downside is, when click-dragging to close the shape, the existing handle moves, though this is the best solution I can hope for."

The last bit depends on how far you are willing to go into silly additions, such as a fake path beginning/guide path. You may, Smart Guides being your friends:

1) With the Line Segment Tool, ClickDrag as you would for the first Anchor Point;

2) With the Pen Tool, Click the end Anchor Point of 1), then the first one to close the guide path and avoid trouble in 3);

3) Start on the real path by ClickDragging the first Anchor Point along 2), thereby exactly (re)creating the desired Handle;

4) Create the rest of the path, and conclude by repeating the ClickDragging from 3) to retain the exact shape/Handle;

5) Delete the guide path 2).

And, maybe less silly, but maybe also less likely successful (at least in the foreseeable future), you may ask for it as a feature:

https://illustrator.uservoice.com/

That is very useful.

The only downside is, when click-dragging to close the shape, the existing handle moves, though this is the best solution I can hope for. Thank-you.

Tommy,

My guess is that you wish to retain the exact original Direction Handle(s) of the first/last Anchor Point without further ado, in other words without your having to redo/adjust what was made already.

Tommy,

"but I'm sure older versions of Illustrator did not do this..."

I am afraid that behaviour is (almost) as old as the hills.