How is it the "market restricting access" if the rents went up because of the eviction moratorium? If that's the reason, then what you said is true in a trivial literal sense, but that's blaming the consequence (market pricing) instead of the real cause (government policy).
Likewise, if housing supply is restricted due to regulations, that's not the "market failing" - that's government policy guaranteeing that rents will be high for poor people.
The only way to make rents cheaper for all poor people is to incentivize massive construction of new apartments, or at least clear all regulatory hurdles to doing so and simply let the market do the rest for you. Flood the market with supply. The more supply, the cheaper rents will be, since demand is mostly stable.
Likewise, if housing supply is restricted due to regulations, that's not the "market failing" - that's government policy guaranteeing that rents will be high for poor people.
The only way to make rents cheaper for all poor people is to incentivize massive construction of new apartments, or at least clear all regulatory hurdles to doing so and simply let the market do the rest for you. Flood the market with supply. The more supply, the cheaper rents will be, since demand is mostly stable.