Replacing default Leaflet L.control.layers with a set of buttons

HTML

<script src="http://cdn.leafletjs.com/leaflet-0.7.5/leaflet.js"></script>
<link rel="stylesheet" href="http://cdn.leafletjs.com/leaflet-0.7.5/leaflet.css">
<script src="http://leaflet.github.io/Leaflet.markercluster/dist/leaflet.markercluster-src.js"></script>
<link rel="stylesheet" href="http://leaflet.github.io/Leaflet.markercluster/dist/MarkerCluster.css">
<link rel="stylesheet" href="http://leaflet.github.io/Leaflet.markercluster/dist/MarkerCluster.Default.css">
<script src="http://code.jquery.com/jquery-2.2.2.min.js"></script>
<script src="https://leaflet.github.io/Leaflet.markercluster/example/realworld.388.js"></script>
<div>
  <select id="mySelector"></select>
</div>
<div id="map"></div>

CSS

html,
body,
#map {
  height: 100%;
  width: 100%;
  padding: 0px;
  margin: 0px;
}

#mySelector {
  position: absolute;
  z-index: 1;
  top: 20px;
  left: 20px;
}

.myButtonBar {
    background: white;
    box-shadow: 0 1px 7px rgba(0, 0, 0, 0.25);
    -webkit-border-radius: 4px;
    border-radius: 4px;
    text-align: center;
    padding: 0px 10px;
    position: relative;
}

.leaflet-verticalcenter {
  position: absolute;
  top: 50%;
  transform: translateY(-50%);
  padding-top: 10px;
}

.leaflet-verticalcenter .leaflet-control {
  margin-bottom: 10px;
}

.leaflet-horizontalcenter {
  position: absolute;
  left: 50%;
  transform: translateX(-50%);
  padding-top: 10px;
}

.leaflet-horizontalcenter .leaflet-control {
  margin-bottom: 10px;
}

.switch-field {
  font-family: "Lucida Grande", Tahoma, Verdana, sans-serif;
  padding: 5px;
	overflow: hidden;
}

.switch-title {
  font-size: 16px;
}

.switch-title:hover {
	cursor: default;
}

.switch-field input {
  display: none;
}

.switch-field label {
  float: left;
}

.switch-field label {
  display: inline-block;
  width: 65px;
  background-color: #f4f4f4;
  color: rgba(0, 0, 0, 0.65);
  font-size: 14px;
  font-weight: normal;
  text-align: center;
  text-shadow: none;
  padding: 2px 12px;
  border: 1px solid rgba(0, 0, 0, 0.2);
  -webkit-box-shadow: inset 0 1px 3px rgba(0, 0, 0, 0.3), 0 1px rgba(255, 255, 255, 0.1);
  box-shadow: inset 0 1px 3px rgba(0, 0, 0, 0.3), 0 1px rgba(255, 255, 255, 0.1);
  -webkit-transition: all 0.1s ease-in-out;
  -moz-transition:    all 0.1s ease-in-out;
  -ms-transition:     all 0.1s ease-in-out;
  -o-transition:      all 0.1s ease-in-out;
  transition:         all 0.1s ease-in-out;
}

.switch-field label:hover {
	cursor: pointer;
}

.switch-field input:checked + label {
  background-color: #444;
  -webkit-box-shadow: none;
  box-shadow: none;
  color: rgba(255, 255, 255, 1);
}

.switch-field label:first-of-type {
  border-radius: 4px 0 0 4px;
}

.switch-field label:last-of-type {
 ...

JavaScript

// Create map
var map = L.map("map", {
  center: [53.905, 27.562],
  zoom: 11,
  maxZoom: 18,
  zoomControl: false
});

// Store for all markers
var allMarkers = L.layerGroup();

// Handle map layeradd event
map.on("layeradd", function(e) {
  if (!e.layer.options) {
    return;
  }

  // Handle only marker layers
  if ((e.layer.options.id != "markerLayer1") && (e.layer.options.id != "markerLayer2") && (e.layer.options.id != "markerLayer3")) {
    return;
  }

  // For the currently added layer (which is one with markers)
  // get all its layers
  var markers = e.layer.getLayers();
  console.clear();

  // Get the dropdown
  var mySelector = $("#mySelector");

  // Sort list items according their ID numbers
  markers.sort(function(a, b) {
    var aId = parseInt(a.feature.properties.id, 10),
      bId = parseInt(b.feature.properties.id, 10);
    return aId < bId ? -1 : aId > bId ? 1 : 0
  }).forEach(function(marker) {
    mySelector.append("<option value='" + L.stamp(marker) + "'>" + marker.feature.properties.id // id added
      + '. ' + marker.feature.properties.name + "</option>");
  })
});

// Handle map layeradd event
map.on("layerremove", function(e) {
  if (!e.layer.options) {
    return;
  }

  // Handle only marker layers
  if ((e.layer.options.id != "markerLayer1") && (e.layer.options.id != "markerLayer2") && (e.layer.options.id != "markerLayer3")) {
    return;
  }

  // For the currently removed layer (which is one with markers)
  // get all its layers
  var markers = e.layer.getLayers();

  // Get the dropdown
  var mySelector = $("#mySelector");

  // Remove options from the dropdown menu
  for (var i = 0; i < markers.length; i++) {
    mySelector.find("option[value='" + L.stamp(markers[i]) + "']").remove();
  }
});

// Zoom on select change
$("#mySelector").change(function(e) {
  var selected = allMarkers.getLayer($(this).val());
  if (markerLayer1.hasLayer(selected)) {
    markerLayer1.zoomToShowLayer(selected, function() {
     ...