This is how to add control icon

[beifeizhou]

Here is how to activate the rectangle and add it as control to the map

const modes = MapboxDraw.modes
  modes.draw_rectangle = DrawRectangle
  const draw = new MapboxDraw({
    displayControlsDefault: false,
    controls: {
      polygon: true,
      trash: true,
      defaultMode: 'draw_rectangle',
    },
    modes: modes,
  })

  map.addControl(draw)
  draw.changeMode('draw_rectangle')

  map._draw = draw

  map.on('draw.create', function (feature) {
    console.log(feature)
  })

Here you can assign draw to a property for map
map._draw = draw

const onClickDrawRt = (map) => {
   map._draw.changeMode('draw_rectangle')
}

<button onClick={() => onClickDrawRt(map)}/>