I just got a secondhand sewing machine. The straight stitch, blind stitch, and stretch stitch settings work, but the zig-zag setting doesn't (the needle doesn't move side-to-side). I plan to use the machine mostly for repairs where hand-sewing would take too long (for example, my first project is a pillowcase that needs a new hem). Is the lack of a zig-zag stitch likely to be a problem?
Based on your planned usage of the machine, it likely won't be an issue for you. The common usages of a zigzag stitch are to allow give in a seam for stretchy fabrics (which you can handle with the stretch stitch setting), to create buttonholes (not a case you described doing), and to secure edges against fraying (which you may encounter, but can compensate for with additional straight stitch lines).
However, I would also like to propose that your zigzag stitch may not be broken, if the blind stitch is working. If you do not have a manual for your machine due to it being secondhand, search using your manufacturer and model name to find a PDF version of it to consult. You may have a stitch width setting which, if turned all the way down, would give the appearance of the zigzag stitch setting not working as the width would be "0."