I'm having trouble understanding your original report, but I will try and answer or clarify some things.

Sometimes the widget appears right next to the edge/vertex you're moving (good!) and sometimes it appears only in the middle of the object (bad, when the object is large and you're trying to position accurately!).

The gizmo (widget) will appear in different places depending on a bunch of settings. For example, when you have multiple objects, edges, or vertices selected, the Gizmo will appear at the median point of everything selected. In edit mode this is the median point of the selected vertices, edges, and faces. In Object mode this is the median of the object origins of the objects you have selected. However, you can change the computed location of the gizmo with the "Transform pivot point" option:
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Note that changing the transform pivot point changes, the pivot point. This is a important thing to keep in mind when using the scale or rotate tool as all actions will occur from the new pivot point defined by that setting.

Create a square plane, go into edit mode, press 2 and select an edge, drag the edge far in one direction using the manipulator widget. Pretty soon you have to move the edge off the screen to actually get to the manipulator to move the edge - which is clearly a nonsensical user interface decision.

This seems unrelated to your original description of the issue. In your original description you talk about how the gizmo is mis-placed when selecting certain objects, but in your "steps to reproduce" you're talking about how you can move a gizmo off the edge of the screen, something different entirely. This also seems like a request to change the current implementation of the gizmo system to stop this behavior. Feature requests should be made on right-click select. Being able to move the gizmo off the edge of the screen is not a bug with Blender.

I should note however that I'm unable to see what's happening on your computer, so the behavior you're experiencing may be a bug. If you would like, you can test Blender 2.83 and 2.90 and see if any of the bug fixes for those versions fix things for you. https://builder.blender.org/download/
You could also share images of what you're experiencing so we can better understand it.

> In edit mode this is the median point of the selected vertices, edges, and faces ...

Not for me.

I’ve attached a screenshot in Edit mode (“1.png") where I’ve selected an edge (indicated by the black arrow). I see no manipulator widget at the median point of the two vertices.

However, if I zoom out a lot I can see the manipulator widget … right in the middle of the object, where I can’t actually use it unless I lose fine control over positioning the edge I’ve selected (see screenshot “2.png”).

As you can see in the screenshots, I have the pivot mode set to “Active Element”. According to the Blender manual: “The active element can be an object, vertex, edge or a face. The active element is the last one to be selected …"

https://docs.blender.org/manual/en/latest/scene_layout/object/editing/transform/control/pivot_point/active_element.html

This is clearly wrong for the edge that I selected to move.

This seems unrelated to your original description of the issue. In your original description you talk about how the gizmo is mis-placed when selecting certain objects, but in your "steps to reproduce" you're talking about how you can move a gizmo off the edge of the screen, something different entirely. This also seems like a request to change the current implementation of the gizmo system to stop this behavior. Feature requests should be made on right-click select. Being able to move the gizmo off the edge of the screen is not a bug with Blender.

I don’t know why you think the widget being unusable due to it’s position “seems unrelated to my original description of the issue”. It’s exactly the same issue - the manipulator widget is not in a sensible place when I’m trying to move an edge.

It’s not a request for a new feature, I’m asking for the existing feature to work as described in the manual.

J.

Can you provide a simple file showing the problem?

Sure, attached.

Select the "lowest" edge, and take a look where the manipulator appears.

The file inded shows an inconsistent status between the selection and the pivot center.

But I can't reproduce the problem from scratch.
The description in the report does not help much.
Knowing how you got there is important to confirm the bug.

I can confirm. But it is difficult to reach the state shown in the blend file.
I edited the description but I wonder if that really makes "the manipulator widget problematic or outright broken in many cases, rendering it useless."

I'm sorry, but if common/fundamental actions trigger the problem (select faces, edges, and vertices in Edit mode, then try to manipulate an edge) then it's not "difficult to reach the state shown in the file".

If the responses in this report are indicative, it's easy to see how bugs in basic functionality would persist in Blender.

@Germano Cavalcante (mano-wii) Looking at the .blend file attached by @john (jgrime) I believe the reason why the gizmo is in the "wrong" position is because the pivot point is set to "active element" but there is no active element. An edge is selected, but it's not an active edge (active edges are signified by it being white) so the gizmo defaults to the object origin.

This is also true for the screenshot shared by @john (jgrime):

Or maybe I'm just mis-understanding this.

Just checking in on this report as it's been a week and a half. @john (jgrime), have you figured out a way to reliably reproduce this issue? Or is the issue due to the "non-active edge" issue I talked about in T76533#927378.