They also leave out the biggest single problem: years of very low interest rates inflating prices, making it hard for people to afford deposits though. There is BoE research showing interest rates are the biggest factor in the earnings to house price ration.
"Immigrants" also leaves something out. Foreign investors and rich foreigners buying holiday homes/second (or third or fourth etc.) homes are a factor in some places (especially London) - in much the same way that people from London and other big cities push up prices in places like Cornwall.