truly zero-metallicity stars

All week I have been discussing with Hans-Walter Rix (MPIA) the possibility that we could find the elusive, truly zero metallicity stars in the Milky Way halo. This from an email I wrote today, reacting to the observational point that no-one has ever seen such a star:

How could there be absolutely ZERO zero-metallicity stars in the Milky Way? Here's everything I got:

1. Maybe there are literally no low-mass stars ever made at zero metallicity. Absolutely none, at high precision. This is possible, given how little we know about star formation and the IMF.
2. Maybe stellar evolution is so weird at zero metallicity that low-mass stars go dark by 13 Gyr. I very much doubt that this is a possibility. But maybe low-mass stars at primordial abundances never burn and just slowly collapse into white-dwarf-like condensed objects on a very long cooling timescale. They would be super cold by now, like 100 K maybe??
3. Maybe low-mass stars form slower than high-mass stars in star-forming regions at zero metallicity. This permits a few of the high-mass stars to quickly evolve and explode, polluting the outsides of the just-forming low-mass stars. These low-mass stars would then have low but non-zero surface metallicities, and be very alpha-enhanced. This is possible, although would it really leave NO zero-metallicity low-mass stars behind?
4. Maybe low-mass stars somehow self-pollute at formation (or later in their lives). Maybe the nuclear fusion kicks in slightly before gravitational steady-state or radiative zones are set up, and the first bit of nuclear burning gets mixed in to the stellar envelopes? These stars would appear to have non-zero (but weird, carbon-enhanced maybe?) abundances. Or maybe this happens later in life because of some weird internal mixing. I have literally no idea whether any of this is possible.