Let objects register to be informed when shuttles have been setup.

* Switch /obj/shuttle_connector to use it instead of the icky process hack.

code/controllers/subsystems/shuttles.dm

@@ -16,10 +16,17 @@ SUBSYSTEM_DEF(shuttles)
 	var/list/shuttles = list()	// Maps shuttle tags to shuttle datums, so that they can be looked up.
 	var/list/process_shuttles = list()	// Simple list of shuttles, for processing
 	var/list/current_run = list() // Shuttles remaining to process this fire() tick
+	var/list/docks_init_callbacks // List of callbacks to run when we finish setting up shuttle docks.
+	var/docks_initialized = FALSE
 
 /datum/controller/subsystem/shuttles/Initialize(timeofday)
 	global.shuttle_controller = src
 	setup_shuttle_docks()
+	for(var/I in docks_init_callbacks)
+		var/datum/callback/cb = I
+		cb.InvokeAsync()
+	LAZYCLEARLIST(docks_init_callbacks)
+	docks_init_callbacks = null
 	return ..()
 
 /datum/controller/subsystem/shuttles/fire(resumed = 0)
@@ -66,3 +73,11 @@ SUBSYSTEM_DEF(shuttles)
 	for(var/obj/machinery/embedded_controller/C in machines)
 		if(istype(C.program, /datum/computer/file/embedded_program/docking))
 			C.program.tag = null //clear the tags, 'cause we don't need 'em anymore
+	docks_initialized = TRUE
+
+// Register a callback that will be invoked once the shuttles have been initialized
+/datum/controller/subsystem/shuttles/proc/OnDocksInitialized(datum/callback/cb)
+	if(!docks_initialized)
+		LAZYADD(docks_init_callbacks, cb)
+	else
+		cb.InvokeAsync()

code/modules/shuttles/shuttles_web.dm

@@ -428,18 +428,13 @@
 /obj/shuttle_connector
 	name = "shuttle connector"
 	var/shuttle_name					//Text name of the shuttle to connect to
-	var/start_time = 60 SECONDS			//After round start (needs to be some time, to let other destinations set up)
 	var/list/destinations				//Make sure this STARTS with a destination that builds a route to one that always exists as an anchor.
 
 /obj/shuttle_connector/initialize()
 	. = ..()
+	SSshuttles.OnDocksInitialized(CALLBACK(src, .proc/setup_routes))
 
-	processing_objects += src
-
-/obj/shuttle_connector/process()
-	if(world.time < start_time)
-		return
-
+/obj/shuttle_connector/proc/setup_routes()
 	if(destinations && shuttle_name)
 		var/datum/shuttle/web_shuttle/ES = shuttle_controller.shuttles[shuttle_name]
 		var/datum/shuttle_web_master/WM = ES.web_master
@@ -452,6 +447,4 @@
 				var/travel_delay = D.routes_to_make[type_to_link]
 				D.link_destinations(WM.get_destination_by_type(type_to_link), D.preferred_interim_area, travel_delay)
 
-	processing_objects -= src
-
 	qdel(src)