> And the argument is: If it is not a difference in pressure then it must
> be gravity that is making the liquid flow.

It's sort of both.
Picture instead the ordinary case of filling a vessel with liquid at atmospheric
pressure through a narrow orifice.
At first there may be a rapid flow of liquid until the headspace gets compressed
and builds up pressure.
THEN there needs to be a chance for the gaseous headspace to bubble up and trade
volume with the liquid for there to be a mass transfer.
The problem lies in only one very narrow tube for bidirectional flow.

If there were a two-way filling mechanism, or a way to vent the gaseous
headspace, then you'd get rapid filling.

Another alternative might be to iteratively overfill with liquid by an increase
in pressure, or by chilling and contracting the headspace, and then letting the
back-pressure bubble up back into the fill vessel.

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