In the zero energy universe model, the gravitational field has negative energy, and this negative gravitational energy of all the distant mass exactly balances and cancels the positive mass-energy in the universe.
Why do we think they exactly balance? If the universe is expanding, the total gravitational potential increases, right? Presumably, we don't live in a privileged time.
A more formal explanation would be preferred, instead of analogies.
First of all, it's extremely difficult to even define the total energy of the universe, and all sorts of different features (dark energy, dark matter, the energy of the cosmic microwave background, the kinetic energy due to heat, etc.) must be taken into account alongside the rest energy of matter and gravitational potential energy. It would be very complicated to do so, while some models may agree with a "zero energy universe", this would be at best a controversial conclusion.
Secondly, you may be overestimating the effect of the expansion of the universe on gravitational potential energy. Gravitational potential energy of a body relative to, say, the Earth, is bounded above by $\frac{1}{2}mv^2$ where $v$ is the Earth's escape velocity. In practical terms, the potential energy of a rock that's 1 light year away from the Earth is indistinguishable from that of a rock that's 10 light years or a million light years away: in all three cases the speed of the rock if it falls to Earth is roughly Earth's escape velocity, and the difference would be far less than any experimental error you could achieve in practice.
