Best thing to do is test it yourself. Without knowing the exact jacket, those straps are usually designed to stop the jacket pulling off your body when you jump/fall into the water
, or if you are too skinny for the jacket to stay on otherwise. So try jumping in from the equivalent height you'd be jumping ship from (depends on your boat) - remembering that you might be falling and unconscious, not able to grip the jacket. Try the worst-case scenario, straight body, hands in the air, etc
You also need the lifejacket to work
when you're lying unconscious in the water. Try wearing the jacket with and without the strap, lying face down in the water and let yourself go limp. You want the jacket to successfully return your head
above water and face up so you can breathe.
There are sample videos of lifejacket tests on you tube.
Having said all that, even if it seems to work
fine without the strap, I would tie it away rather than cut it off. If you ever get in a situation when you really truly need the jacket to survive, you can always call on the extra security
of that strap for some peace of mind. Assuming you're conscious and not too injured to get to it...