Let's assume the CO2 is there, but hidden, rather than missing.

(A near-total lack of CO2 kills your early greenhouse and leads to a frozen White Mars without any fun CO2 flows. Possible, but simple. No water (except rare floods), no oceans, no surface life)

OK, let's think where it could be:

1) Frozen CO2 in the subsurface (dry ice)
- The new results show that the ice of the polecap may get too warm, but maybe OK

2) CO2-H2O Clathrate in the polecap
- Certainly possible, fits temperature data, and leads to more or less the standard White Mars model

3) Pressurised CO2 in the regolith probably liquid).
- Possible, but how do we keep it there and resupply it without a polar basal melt mechanism? So this really only works as an adjunct to (1) or (2)

4) Chemically bound (Carbonate minerals)

This latter is worth some thought. The standard warm wet model for Mars has oceans, or at least rivers and lakes, and postulates a surface sedimentary system where carbonate minerals form and produce nice layers that we will see today, even if buried by later lava etc, due to the many impact craters that act as deep boreholes into Mars' crust.

We don't carbonate in this form, but perhaps we're looking in the wrong place?

Whare we DO see carbonate is in martian meteorites where it formed within the cracks in the rock, some time after they cooled from magma.

A few people have calculated that if the 1-2% carbonate of e.g. ALH84001 were present in the entire regolith, that would consume all Mars' CO2.

So we can have a variant on White Mars where the planet IS frozen, and never (or seldom) has liquid water - EXCEPT at great depth.

Early in Mars history, the surface is still frozen, but carbonate-forming reactions and carbonate precipitation take place at depth in the crust and DO NOT form layered surface deposits.

Any life would also be at depth, and fossils formed there, not at the surface, so the search for ancient biota must go underground (or more easily, seek ejecta from appropriate craters)

In summary, without CO2 freely available, the "fun" bits of White Mars are gone, but the no-water=no-life and frozen aspect of the planet (surface) remain.

Perhaps the observable reduction in size and frequency of flow events is not a progressive freeze of water, but a progressive mineralisation of CO2?

I have more thinking to do, of course, but that's us for now...