I think Wire Cutters removes the wire, but aside from removing any reduction in dice for a unit being in wire, has no effect on whether a unit can battle or not.
So if a unit could move 2 hexes and battle (e.g. Infantry Assault), it could still do so. If it moved 3 hexes, then no battle. If a unit moved 6 hexes using Rush & Rotation, it could cut the wire, and not battle.
In general I think your statement is correct, but there are some exceptions.
"Feed the troops on victory"- Gen. Sir John Monash