This is definitely a personal choice matter. Occasional recreational cyclists tend to eschew any restraints or to use just the clips, without straps. The disadvantages, to me, are the ease with which the foot can slip off the pedals and the total loss of any upstroke torque.

Most folks who regularly spend a fair amount of time in the saddle have migrated to the modern twist-release snap-in binder style pedal-and-shoe system. These take getting used to, and there are numerous stories of bruised egos and other injuries during the learning curve.

A minority of us dinosaurs, myself definitely included, use old school toeclips and straps such as yours, because they work well for us, keeping our feet securely on the pedals and facilitating some upstroke power transmission, while permitting an intuitive jerk-back to release. (Most of us keep the straps just loose enough to permit us to pull our feet back out of the clips -- hard core racers always strapped themselves in and had to loosen the straps to release. Anyone who does that really does need to convert to modern binder-style clipless, for obvious safety reasons.)

My 100% biased advice is to get reacquainted with your toeclip-and-strap system, keeping the straps loose (or absent) at first, then gradually tightening them to the point that you can still pull your foot out when necessary.