That seems to restrict the output probability distributions of all quantum circuits to rather high entropy distributions.

The output of a typical randomly chosen quantum circuit is rather high entropy. That doesn't mean you can't construct circuits that have low entropy outputs (you can), it just means that picking random gates is a bad strategy for achieving that goal.

how can i P(x_i) equal 1/2^n when the bitstrings x_i are sampled from the uniform distribution?

How could it equal anything else? The probabilities of the target distribution have to add up to one, and you're picking each element 1/2^n of the time. For example, if there was a single element with all the probability, you'd score 0 * (2^n - 1)/2^n + 1/2^n = 1/2^n. You always score 1/2^n on average when picking randomly.

How can the value of F_XEB correspond to "the probability that no error has occurred while running the circuit"?

When the paper says "the probability that no error occurs", what it means is "In the systemwide depolarizing error model, which is a decent approximation to the real physical error model at least for random circuits, the linear xeb score corresponds to the probability of sampling from the correct distribution instead of the uniform distribution.".

Physically, it is obviously not the case that either a single error happens or no error happens. For example, every execution of the circuit is going to have some amount of over-rotation or under-rotation error due to imperfect control. But that's all very complicated. To keep things simple you can model the performance of the system as if your errors were from simpler models, such as each gate have a probability of introducing a Pauli error or such as you either sample from the correct distribution or the uniform distribution.

Simplified models actually do a decent job of predicting system performance, particularly on random circuits. For example, consider the way the fidelity decays as the number of qubits and number of layers are increased. The fidelity decay curve from the paper matches what you would predict if every operation had some fixed probability of introducing a Pauli error.

That seems to restrict the output probability distributions of all quantum circuits to rather high entropy distributions.

The output of a typical randomly chosen quantum circuit is rather high entropy. That doesn't mean you can't construct circuits that have low entropy outputs (you can), it just means that picking random gates is a bad strategy for achieving that goal.

how can i $P(x_i)$ equal $\frac{1}{2^n}$ when the bitstrings $x_i$ are sampled from the uniform distribution?

How could it equal anything else? The probabilities of the target distribution have to add up to one, and you're picking each element $\frac{1}{2^n}$ of the time. For example, if there was a single element with all the probability, you'd score $0 \times \frac{2^n - 1}{2^n} + \frac{1}{2^n} = \frac{1}{2^n}$. You always score $\frac{1}{2^n}$ on average when picking randomly.

How can the value of $\mathcal{F}_{XEB}$ correspond to "the probability that no error has occurred while running the circuit"?

When the paper says "the probability that no error occurs", what it means is "In the systemwide depolarizing error model, which is a decent approximation to the real physical error model at least for random circuits, the linear xeb score corresponds to the probability of sampling from the correct distribution instead of the uniform distribution.".

Physically, it is obviously not the case that either a single error happens or no error happens. For example, every execution of the circuit is going to have some amount of over-rotation or under-rotation error due to imperfect control. But that's all very complicated. To keep things simple you can model the performance of the system as if your errors were from simpler models, such as each gate have a probability of introducing a Pauli error or such as you either sample from the correct distribution or the uniform distribution.

Simplified models actually do a decent job of predicting system performance, particularly on random circuits. For example, consider the way the fidelity decays as the number of qubits and number of layers are increased. The fidelity decay curve from the paper matches what you would predict if every operation had some fixed probability of introducing a Pauli error.