What is an acceptable gap between the siding and the wall sheathing?
So I've looked around online quite a bit more since asking this question, trying to see if I can find a building code or some other standard that will give a concrete answer. I didn't find a building code, but I did find some vinyl siding installation guides and, without an explicit building code statement, I would always defer to the OEM installation guides.
The Certainteed installation manual says, on document page 31,

Tips for applying wood furring
If you are working on an older home with noticeably uneven walls,
you must correct this condition before proceeding. If not corrected
during preparation, this uneven surface will produce a wavy
appearance in siding applied over it.
For best results, space horizontal furring strips 12&quot; on center.
Do not exceed 16&quot;. To correct an uneven wall, use furring strips
(and wood shims if necessary) to eliminate low spots.
NOTE: You must apply rigid sheathing over furring.

(Emphasis original). They then go on to give the following graphic (document page 32):

This graphic specifically goes with the installation over concrete or stucco, but clearly shows a sheathing over the furring strips.
The Plygem installation manual says, on document page 17,

Step 4
Install furring in areas needing straightening and leveling. Apply rigid sheathing to  cover and level the furring strips. Do not apply vinyl siding directly to furring strips without sheathing, because the siding may conform around the furred areas causing an uneven appearance.
(See Fig 2)

And then gives Figure 2 as:

Again, clearly showing a sheathing ON TOP of the furring strips. Georgia Pacific siding seems to use the same document (warning: direct download).
The Vinyl Siding Institute's 2018 installation guide seems to be a little more ambiguous, stating on document page 18,

Make sure that any furring strips are thick enough to provide this penetration depth, or cover them with wood sheathing to provide the needed depth.

The &quot;or&quot; there is what makes it ambiguous to me, but the top of that page also clearly states:

A flat, level wall surface is necessary for proper installation of vinyl siding.

They also give the following graphic on document page 15:

So, ultimately, I think the decision here would be to use standard 1x3 furring strips (NOT pressure treated), such that they're dimensionally 0.75&quot; thick, and then to face those with 0.5&quot; foam insulation. Some reviews have said the furring strips are actually 5/8 thick, not 3/4, but it should be close enough for the siding to fit into the window trim (though I'd definitely dry fit one section before investing in a whole house's worth.)
Finally, regarding the NOT pressure treated furring strip, the general practice of installing the furring strips to create a cavity seems to be referred to as creating a rainscreen, and the author of that article makes a comment saying,

Using pressure-treated lumber is not only unnecessary -- it often requires the use of expensive stainless-steel fasteners.
Lots of researchers and ordinary builders have removed siding and inspected furring strips. The furring strips stay dry.
If a wall has rotting furring strips, something is seriously wrong with the wall design or installation. Rotting furring strips would usually be a sign of a serious flashing error.

Finally, I'll note that the article and the video I linked in the question all seem to be using furring strips without a sheathing as part of a wood siding installation. Again, all the sources I could find on vinyl siding are OEM installation guides and they're all uniform in their statement:
You must install vinyl siding on a flat surface. Attach sheathing to furring strips if you need to use furring strips.