You are welcome.
I think I can close this with your shorter answer as a TLDR recipe :)

Got it - thanks Egor! The simple answer is: don't use the Target column, use the Selection column. (Doh! I knew that.)

Thanks for your patience!

Although it looks like it selects all objects in a layer, it does not do that. It targets the layer contents, that is the layer itself.

Try to target a layer by clicking this circle marker and then apply an effect to it, something simple, like Effect > Path > Offset Path (but please check it on a simple enough layer).

What you get is this effect applied to the layer itself. If you add a new object in it — it will get the offset too, although it was not selected before, right?

If you want to select all objects in a layer, you should do the other thing. Undo this experiment with Offset. Deselect all. Instead of clicking the circle marker for the layer, click the square marker next to it instead. This is what selects all the content of a layer in Illustrator. If you do that, you will be able to create an Intertwine just fine.

Also, try to hover over the circle and the square marker to read the descriptions they have. Circle one has 'Click to target, drag to move appearance'. Square one has 'Indicates Selected Art (click to select art)'.

If you refer to the manual (https://helpx.adobe.com/illustrator/using/layers.html), it says:
'Target column indicates whether items are targeted for application of effects and edit attributes in the Appearance panel.'
So these markers target, to work with the appearance. The Appearance panel even has the icon with this marker. Intertwine is not an appearance in Ai, it’s a wrapper. So when you target a layer, you try to intertwine the layer itself, that is put the layer inside of the Intertwine object — which is impossible, since a layer can’t be put in object.

Hi Egor,

Clicking the target indicator does appear to select all of the objects on a layer—you can apply fill/stroke to all the selected objects as well as perform actions such as rotate/scale.

This leads me to expect that Intertwine should recognize the objects as selected.

Patti, this is an expected behavior.
See, when when you target a layer by clicking the dot marker next to its name in Layers, you do not 'Select All' in a layer, you actually target a layer itself, as an entity, to be able to apply effects to the chosen layer, and not the objects in it.
Thus, when you target a layer, you try to intertwine the layer — and it should not an won’t work for a wholesome layer.
Given this — do you still think it should be possible?