Laws and regulations often limit or restrict remedies in various ways that may not be intuitive or correspond to popular ideas of legal rights.
In the case of the recent US eviction moratorium, the obligation to pay rent was not changed. The only effect is that no action for eviction could be brought for a residential property during the effective period of the order, in situation to which the ordeer applied.
The order says (see link above):
This Order does not relieve any individual of any obligation to pay rent, make a housing payment, or comply with any other obligation that the individual may have under a tenancy, lease, or similar contract. Nothing in this Order precludes the charging or collecting of fees, penalties, or interest as a result of the failure to pay rent or other housing payment on a timely basis, under the terms of any applicable contract.
To invoke the order a resident must sign a statement declaring under penalty of perjury that:
- I am unable to pay my full rent or make a full housing payment due to substantial loss of household income, loss of compensable hours of work or wages, lay-offs, or extraordinary  out-of-pocket medical expenses;
- I am using best efforts to make timely partial payments that are as close to the full payment as the individual's circumstances may permit, taking into account other nondiscretionary expenses;
- If evicted I would likely become homeless, need to move into a homeless shelter, or need to move into a new residence shared by other people who live in close quarters because I have no other available housing options.
as well as various other declarations.
The order's justification section says in part:
Evicted renters must move, which leads to multiple outcomes that increase the risk of COVID-19 spread. Specifically, many evicted renters move into close quarters in shared housing or other congregate settings
The order later specifies that:
This Order shall be enforced by Federal authorities and cooperating State and local authorities through the provisions of 18 U.S.C. 3559, 3571; 42 U.S.C. 243, 268, 271; and 42 CFR 70.18. However, this Order has no effect on the contractual obligations of renters to pay rent and shall not preclude charging or collecting fees, penalties, or interest as a result of the failure to pay rent or other housing payment on a timely basis, under the terms of any applicable contract. ...
While this order to prevent eviction is effectuated to protect the public health, the States and units of local government are reminded that the Federal Government has deployed unprecedented resources to address the pandemic, including housing assistance.
It would seem that civil suits for payment of debts arising from unpaid rent are in no way prevented, only a remedy of eviction is blocked. That block would apply to any state or local courts, as well as any federal courts. Eviction cases are not normally brought in federal courts. But seizure of bank accounts or garnishment of wages could be obtained, if permitted by state law in a particular case. So might other remedies authorized by state or federal law.