Randomly overheats. New thermostat,coolant topped off,no leaks, and no oil or gas in the coolant .
high mileage? Might be a worn water pump. Pop the radiator cap and see how well coolant is flowing. Old radiators can get gunked up too, to the point little coolant is flowing through them. An IR Thermometer can check is entire radiator is working or if there are "dead" spots. After that, try giving it a cooling system pressure test, run Block Tester, then a compression test (head gasket).
Your coolant pipes might be blocked but even if they are working you light even have a faulty fan in my case the connector was broken and I almost seized my engine check that the fan comes on when the engine gets hot if the fans come on then it might be the water pump as KenF suggested or that the pipes are blocked
In our I6 engine platform, the most common problem would be the temperature sensor. Our gauges are HORRIBLE!! Hook up a scanner and see the real temp next time it "overheats". 2nd, would be fan clutch, third waterpump. Try to wiggle the fan back and forth, should be no movement. You didn't say which year of Trailblazer for locations of temp sensor. 02-05 are located right above the thermostat on drivers side, easy. 06-09 is located on passenger side, way in the back near firewall, PIA to get to.