While the coolant temp gauge is not linear and is not always 100% accurate, especially if the stepper motors have ever been replaced, I have found over the years that the gauge reading for most people is a fairly accurate indicator as to the health of the thermostat, and hence the cat. This is why that was my first question when I read your symptoms.
With some exceptions, I would still tend to say that a temp gauge left of center means the thermostat is likely bad. If it has been like that for a while, then the cat is probably plugged. Cats run on a very narrow range of mixtures, and as these platforms age, we are going to see more and more plugged cats. If the driver has also had a few misfires in the past or if the vehicle has ever been driven even just a few miles with a check engine light flashing, the cat is almost certainly bad.
A plugged cat will not throw a trouble code.
Testing will confirm this. Coolant temperature on a scan tool needs to be 205 to 210 when warm. If it is anything below 190, then your sensor is bad or the thermostat is bad. If it has been like this for a long time, the cat is almost certainly plugged.
The cat can be tested with an exhaust backpressure test. They pull out the front oxygen sensor and replace it with an exhaust backpressure gauge. Backpressure at idle should be 0 PSI and no more than 1 to 1.5 PSI at 2500 RPM.
This is also a good time to replace the front oxygen sensor. We recommend a new sensor with every plug change, if for no other reason than to prevent them seizing in the manifold after many years.
Don't let the shop test the cat by just feeling it or reading the temperature. A backpressure test is definitive. (I own my own exhaust backpressure gauge and an entire set of oxygen sensor sockets because we have two Trailblazers in our family and that's how sure I am that your cat is partially plugged.)