Yes, attenuators are what you’re missing, also known as matching transformers. You can get them as small self-contained units that look like mic couplers, Shure makes a couple nice little ones, some switchable in level.
Typically PA systems and mixers don’t have a lot of extra XLR outputs to feed you, and almost never a spare mic-level out, you usually have to settle for line level thru a phono jack or RCA jacks. Balanced audio all the way thru is what we prefer, and radio shack adapter voodoo at the back of the board may introduce sources for hum and the like. Something to watch out for.
In desperate situations, I have also gotten acceptible results putting a dynamic stick mic or even a wireless lav next to a PA speaker. Hey, they mic electric guitar speaker cabinets and the like, so it’s not so weird, it’s just not usually the best practice in video of events, because any buzz or hum in the PA gets magnified. If you place the mic far from the speaker, enough to reduce the buzz, you get audience talking leaking into the feed.
One other consideration is the difference between a “wet” and “dry” mix, and the difference in a mix made for PA versus one made for video. The mixer guy may have EQ and such tweaked to best deal with the room acoustics for the audience’s ears, and that may not be the same as mixing for good video use.
You also need to worry about if you need audience response and ambience in the tracks somewhere… reactions to jokes, applause, etc. which the PA feed may or may not give you. Shotgun on a second channel on the camera may or may not be enough for that.
My preference is to run a hardwired stick mic to the podium and use good shielded cables all the way back to the camera. Everything else besides that is a compromise forced by the specific circumstances.