Premiere Pro treats my S-Log3 footage as Rec.709. What am I doing wrong?

Are you working in the Lumetri panel's Settings tab ... where the entire looooong chain of options is available?

If so, look up and down the list. You probably don't want auto=detect log on, especially in companion with a Rec.709 sequence, as it will do a full algorithmic transform to Rec.709.

Also, input tonemapping on the Sequence setting options should be off for you.

Their algorithmic transforms were designed in tandem with the camera makers, and are a good replacement for the rather more simplistic ... and problematic ... LUT transforms.

LUTs are just a basic look-up table ... they have like 32 o4 64 entries of 'this RGB triplicate changes to this RGB triplicate" ... and in-between, the app just applies a normally pretty straight line transform. No shaping or anything.

And LUTs are called the dumbest math out there by colorists for that reason. And they will crush and clip data if the field produced media does not quite closely match the exposure/contrast/saturation of the media used to create the LUT to begin with. 

I work for/with/teach pro colorists, and they tend to all go for algo-based transforms when they can. Safer for your pixels, is how several describe them. A slightly different starting place than any particular LUT you may have been using, but then, most makers have a wide selection of provided LUTs for each camera anyway. They are also only a starting point, like the algos.

Now ... to do a full pass through, without any changes, hit the "preserve RGB" option up top. At that point, if you have log-encoded media, and say do a ProRes output, that will be the same log-encoded data.

And ... if and when using their rather spiffy new CM setup, one thing you may have hit is to not have sequence input tonemapping on if you want to work in the wide gamut ACESScct working space and do your own normalization.