How to toggle the visibility of the starting layer in leaflet.js file

After declaring your layers, you need to add it to the map, just like you did with your playlist. See code below. This should work. The control layer will work to keep the layer visible / added to the map and check it in the list.

var map = L.map('map').setView([52.814172, -2.079479], 9);

L.tileLayer('http://{s}.tile.stamen.com/toner/{z}/{x}/{y}.jpg', {
    maxZoom: 18,
    attribution: 'Map tiles by <a href="http://stamen.com">Stamen Design</a>, under <a href="http://creativecommons.org/licenses/by/3.0">CC BY 3.0</a>. Data by <a href="http://openstreetmap.org">OpenStreetMap</a>, under <a href="http://creativecommons.org/licenses/by-sa/3.0">CC BY SA</a>.'
}).addTo(map);

// Left out layer-declarations, they are HUGE

anti_soc.addTo(map); // here it is

var overlayMaps = {
    "Anti-social behaviour" :anti_soc,
    "Criminal damage and arson": crim_dam,
    "Burglary": burg, 
    "Violence and sexual offences": viol,
    "Other theft": other_theft,
    "Vehicle crime": veh_crime,
    "Shoplifting": shoplift,
    "Public order": pub_ord, 
    "Robbery": robb,
    "Bicycle theft": bikes,
    "Drugs": drugs,
    "Theft from the person": theft_person,
    "Other crime": other_crime,
    "Possession of weapons": weap
};

L.control.layers(null, overlayMaps).addTo(map);