"we know that cold air mass sinks while warm air mass rises due to density difference, resulting in higher pressure in cold air mass"

Wrong. Only hot enough air rises, not all hot air. It's hot enough when the entropy/potential temperature profile has a negative slope w.r.t altitude.

Your pressure in the cold region is lower than in the hot region, same as for the thermal wind. But this is due to different balancing requirements (vertical vs horizontal), so you're not comparing apples to apples here.

Does this contradict the concept of monsoons? No because monsoons are surface-level winds. Let's use land and sea breezes as an example since they are basically just small-scale monsoons. Take a look at this image regarding land and sea breezes. Air in the warm region rises which causes the isobars to "spread apart". A horizontal pressure gradient in the upper troposphere occurs so wind will flow from the warm area to the cooler area; this is thermal wind. Air above the cooler area sinks because it is more dense. This gives room for the thermal wind to also sink above the cooler area. The surface pressure in the cooler area is now higher than the warmer area since the sinking air is building up. A horizontal pressure gradient forms which causes wind on the surface to flow from the cooler area to the warmer area.

I hope you understand since explaining this was a bit tricky without a lot of visualization. Also, check out wind flow maps of a monsoon. On the surface, wind flows from the colder area to the hotter area. But if you check the wind flow at higher altitudes, the wind actually reverses.

Edit: Here's an analogy I just thought of. Think of the air as individual molecules. Think of these molecules as balls in an aquarium. These balls would always want to be a set distance away from other balls. If a ball is too near to another ball, they will push each other; if a ball is too far from another ball, they will try to go closer.

One end of the aquarium is hot while the other is cold. The balls in the hot part rise and build up at the upper portion of the aquarium. The balls in the cold part sink and build up at the bottom part of the aquarium. The balls build up but since they don't want to be together, they flow towards the spaces with no balls. The hot balls will glide on the ceiling of the aquarium towards the colder part where they cool down and sink. The cold balls will glide on the floor of the aquarium towards the hot part where they heat up and rise. The cycle then just keeps repeating.

The thermal wind phenomenon occurs on the ceiling of the aquarium, the monsoon/land-sea breeze phenomenon occurs on the floor of the aquarium.