I like both twine and wire, but like wire a tad bit better.
You don’t necessarily need wire cutter if you have a wired bale. If you look at the corner of the bale, you will find where the wire was twisted to close from the baler, and just have to un twist the wire to get the bale opened (great fall back if you mis place the wire cutters or just don’t have any hand). If you find the twisted wire on one side of the bale, the other side is generally the same place on the other wire.
Wire bales also stay formed longer if you open a bale, only use a few flakes and then have to close the bale and save for later. The twin just doesn’t keep the bale as neat and tidy as the wire does when you have to close the bale after you pull a few flakes from it. Twine is handy to have around, but at times so is wire.
If you are worried about the sharp points of cut wire, just bend the ends down into the hay… or back on themselves.
And, if I’m moving a lot of wire bound hay, just use a pair of gloves and your hands will stay fine, twine just feels (to me) like it is going to break and loosens up on the bale when moving it from stacking in truck to truck to storage area.