I wouldn't say it makes me safer directly, but it makes me less stressed which in turn removes the pressure for the 'daily race' which does make me take less risks, which in turn is safer I suppose.
Before HM I would want to be first off a motorway so I didn't get 'stuck' behind the slower car, truck, van etc. So I would pull a typical late lane change or overtake to make sure I was ahead. Nothing 'dangerous' or so I thought but looking back there have been some hairy moments but of course we are all driving 'gods' or "The Stig" are we not ?
Now I don't care, in fact I look for a slow truck or bus to follow on most motorways because it means it isn't me in my small car causing a lane to be blocked, its that large, visible truck in front of me. This helps a lot as most Scottish motorways only have 2 lanes - did you read about the "tight" Scotsman
Not needing to beat someone from the lights removes a lot of stress and being slower means more looking ahead. Quite a lot of the time I arrive at a roundabout last in a queue but because I have judged the traffic well I enter and exit it first.