ESP-WROOM-32 layout USB-UART
Do you need RTS? Yes and no. Per the reference schematics for the ESP-WROOM-32, the RTS pin is needed (along with DTR) if you want the programming process to be able to automatically restart and put the ESP device into the correct "Boot" mode for programming and then automatically restart it and run in the regular boot mode after it's done programming. If you don't have the RTS pin, then you'll have to use a separate mechanism (like a pushbutton) connected directly to the ESP-WROOM-32 pin IO0 so that you can power it up in the correct boot mode for programming.

I am somewhat familiar with this process, as I have split off the ESP programmer / power circuit into my own custom PCB, following the same reference schematics that you're looking at.