Its not the easy access to guns that makes it so rare. Most home invasions in the USA are done by people without guns, largely because if they are robbing your home they are like drug addicted and/or poor and never had the money for a gun.
The reason home invasions in the UK is rare is likely many other factors such as how children are raised or easier access to mental health.
In fact when the UK banned guns violent crimes of all sorts skyrocketed and never really recovered. So from a relative perspective this seems contrary to the facts.