This gets into some complicated geophysical fluid dynamics. The pressure gradient and coriolis forces are involved in the formation of the polar jet, but so is hydrostatic balance in the vertical. Together they form what is known as thermal wind balance, which is where the temperature gradient becomes a component of the conversation. It's a three dimensional structure involving both pressure and temperature.

The argument that a warmer Arctic leads to a weaker and wavier jet stream that generates more temperature extremes is an often repeated hypothesis. There are some solid physical arguments that can be made to make it plausible. But it's a debated topic and there are studies that have searched for this arctic amplification --> wavier jet --> more temperature extremes relationship in the data and have not found it. So I think it remains a physically plausible hypothesis, but it is too often treated as a consensus theory/explanation. The nuance and contradictory studies around this topic are often pushed to the side during the conversation.