Is a sway bar really that important on a stock,no frills,zero performance oriented car such as a station wagon?

One of the sway bar brackets on the frame of my 72 Fury Wagon has broken the welds.

If I choose to take the sway bar off the front,will I notice any difference in handling? Im not running an Autocross or any other slalom type event. Just normal everday driving in traffic and such.

Is it safe to remove it?