How do people with Macs set up data sources in ODBC on a Mac to connect to a PostgreSQL database? What do you download and install so that the "Connections" tab in RStudio can connect to a PostgreSQL database?
Have you seen the documentation at https://db.rstudio.com? Specifically, the following pages might help with your question:
https://db.rstudio.com/rstudio/connections/
https://db.rstudio.com/best-practices/drivers/
Yeah - I found that after I posted the question. It's not linked from anywhere obvious in RStudio and Google didn't turn it up.
I found this blogpost to be very useful while setting up Postgres on Linux, but seeing that MacOS is still Unix-based, I believe it was working for Mac as well.
What I also found is that ODBC didn't work quite well with Postgres. Specifically, I had an issue that all long character columns were being cut off after 512 (or 1024, I don't remember) characters. This didn't happen with RPostgres. One more thing that kept happening with ODBC is that I would quite regularly click somewhere on "Connections" pane and then R/RStudio would try to load gigantic table in memory and eventually crash. This, of course, wouldn't happen with RPostgres for the simple reason that there is nothing to click 
But having "Connections" pane is really useful, I agree, so maybe for you those drawbacks are not that important.