House isn't getting down to the temperature that I set.

I purchased a house with a central air system that wasn't working. During the renovation, I completely gutted the house, and the HVAC company was involved from the very beginning while the walls were still open, giving them easy access to install everything properly.

I purchased an entirely new HVAC system, and it took them about a week to complete the installation. When they finished, they told me everything was operating correctly.

We continued with the renovations, and now that the project is nearly complete and the weather has warmed up, we've discovered that all of the bedrooms are significantly hotter than the rest of the house.

I had the HVAC company come back to inspect the system, and they told me the existing ductwork was inadequate and needed to be replaced. They also pointed out that the master bedroom only had one supply register. They used an airflow calculation app and claimed one register was sufficient, but I insisted they add a second one, which they eventually did. I don't understand why this wasn't identified before the walls were closed, because now my contractor has to repair the drywall they had to cut open.

The house now has all new ductwork, but the problem still hasn't been resolved.

The thermostat is located in the open-concept living room, dining room, and kitchen, which has four supply vents. Even with the thermostat set to 68°F, that area can only get down to about 70°F. Meanwhile, the master bedroom stays around 73°F, and the other two bedrooms are about the same temperature.

I had them come back for a third visit, and now they're recommending adding an inline blower/booster fan to the ductwork to push more air to the bedrooms.

My concern is that even the main living area—with four supply vents and the thermostat—isn't reaching the temperature I set. If that part of the house can't cool to 68°F, it makes me wonder if the issue is more than just airflow. It almost seems like the system isn't producing air that's cold enough, but I'm not an HVAC expert.

What do you all think? Does adding a booster fan sound like the right solution, or does it seem like there's a larger issue that should be addressed first?

The house is a little shy of 1800 sq ft. I'd say the LR, dining, and kitchen is around 600-800 sq ft.