Thank you for moving this to the Renting forum from the Austin forum.
A landlord does not have to work through a real estate agent or leasing agent, they can do it all themselves.
If you ask a real estate agent to help you find someplace to rent, they will probably charge you a fee.
Practices may vary from place to place, but in Austin there are numerous apartment locators who will help you find an apartment or a house and they do not charge the tenant, they are paid by the Landlord. Some landlords have leasing agents and property managers working for them, many do not.
Once you have connected with a landlord or leasing agent, most will expect you to pay a non-refundable application fee which covers their costs of doing a credit check and sometimes a background check on you to verify if they think you might be a good prospective tenant.
I have never used a real estate agent to find a tenant for me so I do not know what they charge, but it is probably a one time charge.
After that it is up to the landlord or his property manager to deal with the tenant.
A leasing agent and property manager are often the same, but it is certainly possible that one could have a leasing agent who handles only the initial lease, then a property manager does the rest. Or the landlord does it all themselves.
There is a lot of good information available from the Austin Tenant's Council here:
Austin Tenants’ Council/Tenant-Landlord Fact Sheets
You can read more about the Texas landlord-tenant laws from the links in the Sticky thread at the top of this Renting forum.