Yup Beatty did complain and I'm sure he used that good old Panzer division tactic of fabricating operational strenght to avoid losing more stuff to other theatres. In 1917 the Germans had opened up the U boat war which was draining RN destroyers, but was this a factor in 1914/15?
..... There is an interesting book, "Keeping the Seas" by Captain E R G R Evans, who commanded 6th DD Flotilla out of Dover from 1914. Evan writes that due to the submarine threat as early as October 1914 "orders came down from the Admiralty that no vessel except for a destroyer, or a scout at full speed was to cross the Channel during daylight hours. <snip> the patrolling cruisers were withdrawn, and the destroyers had to be organized to patrol the areas which has been formerly watched over by the four light cruisers or scouts ofthe ATTENTIVE class; it meant more patrolling than ever." Evans mentions further that any submarine sighting caused the despatch of a division of DDs to hunt it, a practice which was no doubt pursued elsewhere along the North Sea coast as well.
A close reading of Jellicoe's "The Grand Fleet" supports this surmise, as he mentions destroyer divisions forever being sent off here and there to patrol for submarines or in reaction to any submarine sightings. In Jauary 1915, the GF officially possessed 40 destroyers attached in two flotillas [2nd and 4th] of 20 boats each; of those 40 boats, 10 were re-fitting and 8 were on detached service in the Irish Sea, leaving only 22 boats for fleet duty.
Probably the most telling point with respect to the demand for destroyers for patrol and light force interdiction duties was the size of the Harwich Force [30+ DDs] and the fact that they had first call on the most modern destroyers.
For timing you have to have good intelegence which is what the Germans simply did not have. The only other way to find the weakness was by continual probing which inturn would lead to the servicability going down. So you could have too choices 1/ Go in with high servicabilty no inteligence and little practice and hope the RN was having a understrength day or 2/ Continually probe and if the response seems weaker than usual hit them with your fleet but have a lower sevicabilty. Both have a certain amount of risk.
..... I think that approaches the matter with a bit too much hindsight. Jellicoe had no way of knowing the quality of German intelligence at the time and it was his obligation as the responsible person to plan for worst case contingencies.
Yup I know they were from Harwich but Im sure Jellicoe kept a couple of destroyers with him to escort his deradnoughts home.
..... IIRC, Jellicoe had four flotillas with him at the time and surrendered two to escort LION.