One issue with this type of system is, who defines what the truth is? If it's the government in power, then it'll flipflop immediately whenever the other side wins. Any sort of "independent bureaucracy" would also gradually be undermined.

This is also ignoring that lying in the house of commons is likely not a major cause of political polarization, given that I imagine very few people actually watch the sessions of parliament. I would attribute the rise of political extremism mainly to decreasing quality of life, which arises from a combination of the housing crisis, heinous wealth inequality, and the looming specter of climate change. After all, if the status quo doesn't work for the average person, they will naturally look for alternatives (or get politically disengaged). The right wing has the edge here, as they have a multi-billion dollar propaganda complex that is very effective at getting their word out, and they have no end of scapegoats to blame. The left has no such network, but we do have the benefit that most people aren't massive racist assholes, and it doesn't hurt that the actual facts back us up.

It is also worth noting that this type of polarization cannot be avoided under the system of capitalism, which broadly pits two groups against each other: the working class and the capitalist class. The working class broadly seek to live comfortably, to receive as much money as possible in pay, and to work no more than is necessary. The interests of the capitalist class run directly counter to this: they seek ever-increasing profits, and would like to pay the working class as little as possible and have them work as long as possible in exchange. This adversarial relationship can be overcome temporarily by a social contract that enacts high taxes on the capitalists to pay for a welfare state (note that even this arrangement does require exporting suffering to the global south through the mechanisms of imperialism, which I'm going to ignore for brevity), but eventually the drive for ever greater profits will drive the capitalists to destroy that social contract. Two relatively obvious examples of this are that wages have become stagnant for the last 50 years despite massive increases in productivity, and the movement to destroy public healthcare.

In sum: if we are to eliminate polarization, we must first eliminate capitalism. 😊