To answer your questions:

"...why anyone would work for an agency?"

Using an agency allows a lady to just show up for appointments and still have a life outside of the hobby.  Being an independent is pretty much a full time job, and requires commitment to do it well.

"...would it not make more sense for her to handle that herself?"

Some ladies do not have the necessary time, discipline, desire, patience, and/or aptitude to run their own business as an independent and handle the day-to-day and hour-to-hour responsibilities.

"Is there really any advantage to providers OR clients in using an agency?"

For providers, see above.  For clients, it's mostly convenience.  There is usually significantly less advanced notice required to get an appointment with an agency.  Once screened by an agency, the client does not have to undergo additional screening to see a new or different provider listed with the agency.

Some cities, working for an agency is a lot easier since Vice may be a lot more aggressive.

Other cities, it's more of a stigma to work for an agency and independents are more the rule for being good like here in LA. There are very few agencies that are very reliable with great rosters
here in LA.

RH

There does seem to be differences among cities as to which is better. I imagine some of it has to do with in which environment one feels more safe.