You can try to use these scripts: https://onthehead.com/ais/preference006/
Still you have to find a way to assign them to the hotkeys you want, but it's better then nothing

I was able to set this up for Windows using Autohotkey script. Shift+Numpad key does this for me.
Still this need to be done from the box.

You can do this now by creating an action for each point then assigning them f-key shortcuts. You have to select the object first, of course.

I have a Razer Tartarus Keypad at both work and home, so I assigned each shortcut to a keymap, so it's a one-button press function and I don't have to remember the function key or whether or not I need to hit CTRL+SHIFT, etc.

Absolutely. Keyboard shortcut keys for reference point selection is something I'd nearly pay for. Hang on, we are... paying any way... I use a shortcut keyboard for reference point selection nearly 40 times a day in InDesign. It's a nobrainer that it's need in Illustrator.

This is a great idea. Vote for it here, it has only 7 voices right now:
https://illustrator.uservoice.com/forums/333657-illustrator-feature-requests/suggestions/32403805-make-setting-reference-point-hotkeyable
Also voted here.

This would be such a time-saver!
And InDesign also has this feature. (Strg + 6, then Shift Tab, then numeric keypad) but there the reference point runs out of focus after setting once.
I just imagine pressing "1" of the numeric keypad while drawing a rectangle, the right hand always on the mouse. Make a copy of the object (Strg + C; Strg + F) and just type on "6" of the numeric keypad to change the reference point

God. That is such a great idea.

If I select a shape on the artboard that I want to enlarge while keeping the top left corner in place, I mouse-click the proxy indicator (attached) before the shape modification event.

I wish with all my heart, you could provide key commands for this. It could be something like ALT + X + 1, 2, 3, 4, 5, 6, 7. 8. 9 on the keypad. By selecting number ALT + X + 4 would position the proxy indicator at middle left, to mirror the layout of the number pad.

Well I've been keeping an eye on this site and you do seem to post more than anyone else :D

Amazingly, I don't think I actually disagree with anything you've said and often your comments are spot on.

From the way in which features have been added to, and neglected, in Illustrator, it appears to me as though they're lacking someone with a detailed real-world understanding of how these tools are used - and that's exactly what we, as customers, really need them to have.

Mark, thank you for your generous appreciation of my humble efforts. How could you notice that? :)

Thinking more about 9-grid, I see no possible interference with pivot-based tools yet (Scale-Shear-Rotate-Reflect).
There should be definitely visible distinguishment between locked-fixed pivot and normal one. And I guess it could be toggled just by clicking it twice. When it's locked, it could stay visible even when not-pivot-based tools are active (but there should be an option to hide it).

Interesting approach, guess it depends what you're making. For regular 2D shapes I can definitely see why you'd want to be able to hot-key the reference point position. I tend to just flip about the vertical axis (with an action, as you say).

RE persistent pivot point: you've really thought this through, I'm impressed, I agree with your specification. How will it work with the 9-point system? I don't know, you'll come up with a way :p

ADOBE: Your customers need you to hire this guy. He knows Illustrator inside out, has an eye for detail and thinks things through. As a customer, I'd be a lot more confident in Adobe's future potential to meet my team's needs if I knew someone like this was looking after the tools in Illustrator.

Mark, I personally use reference point for mirroring and rotating (and setting coordinates, of course). I mostly use keyboard rather than mouse and live tools to do this, because of my workflow. I just press F4/Shift+F4 to mirror instantly, using Actions. Sometimes I need to flip over top-left point, for example, and next time — from bottom-center, and do this very often.

As for your idea for persistent pivot point... I don't know.
This seem to be a good feature, I think I saw people request it, it surely need to be requested if not, but it raises many questions. Most of them are easily answered, but everyone has his own truth. AI team had troubles with 'Zoom to selection' feature, which was one thing for me and other things for everyone else, including Adobe's vision.
What will happen if you set a custom pivot for a group and the ungroup it? Will parts of it adopt this pivot? (I would say no).
Will new group respect pivots of parts? (no)
Should be there a quick way to reset a pivot to center? (YES)
Should there be a way to move object while pivot stays? (NO — it would give more mess than benefit)
How this persistent pivot will correlate with current reference 9-point-grid?

Hmm - how would it work? For me, hotkeys for sending the reference point up/down/left/right wouldn't be that helpful. However, simply having a key that picks up the reference point and sets it down wherever you click would be great. Even more so if that reference point actually saved with the object for the next transformation.

The only way to set reference point in AI — is to click precisely in Reference grid in Control panel/Transform palette.
I rely hugely on keyboard in my work, and I'd like to set points via hotkeys.
But while Indesign allow accords to do it (https://indesignsecrets.com/keyboard-shortcuts-to-the-reference-points.php), AI does not (https://forums.adobe.com/thread/1466926)
Please do something about it. Numpad would be so nice to use it for this task.