Using a broker to find an apartment to rent is a very alien thought to me but I guess I've only ever really rented in big apartment complexes. I would normally just browse the area I wanted to live in on Google/Bing maps, find a few places that looked interesting, see floorplans on their websites. Take the top few of those and spend a Saturday driving to each of those to check them out. I guess if I was trying to find a place with a lot of independently owned apartment units you'd need a broker to find stuff, but really it seems like something that doesn't need a broker getting paid several hundred dollars for an evening and a day of inconvenience of shopping around. I mean, you're probably going to spend that Saturday viewing the apartments anyways, now you just have someone you're paying to join you.
Lease contracts in my state are pretty much entirely standardized. Pretty much every place uses the same lease that has a bunch of fill in the blanks for amounts, unit numbers, etc. There's not a lot of additional forms to be filed. When I bought a house I was happy to have a real estate agent with me as there were a lot of forms, several different 3rd parties to deal with, much more risk, and the whole process was a lot longer. Plus you pretty much need an agent to get in to the more accurate MLS listings. There would be so many homes still listed as for sale on sites like Zillow and others that were already sold while the MLS listings were usually up to date within several hours.
Brokers are typically hired by the landlord to find a tenant and not a tenant to find an apartment. The big apartment complexes with a company ownership rather than individual typically hires brokers too, but probably they have enough apartments to just have in-house ones and pay them themselves.
But at least in NYC those big apartment complexes are typically only at the higher end of the spectrum. You'll be using a website like StreetEasy to find listings online which often don't have the best pictures, floor plans and you will have to schedule an appointment with one such broker for a time that works best for both of you. These days its less common but pre-pandemic it was not uncommon for a tenant to have to pay 1 month rent as broker fees.
Lease contracts in my state are pretty much entirely standardized. Pretty much every place uses the same lease that has a bunch of fill in the blanks for amounts, unit numbers, etc. There's not a lot of additional forms to be filed. When I bought a house I was happy to have a real estate agent with me as there were a lot of forms, several different 3rd parties to deal with, much more risk, and the whole process was a lot longer. Plus you pretty much need an agent to get in to the more accurate MLS listings. There would be so many homes still listed as for sale on sites like Zillow and others that were already sold while the MLS listings were usually up to date within several hours.