Would it not be fair to suggest that submarines were something of an unknown quantity, even by 1914, many dismissed them as something of an irrelevance.
I get the impression that most naval personnel took submarines quite seriously by the start WW1, in something of a repitition of the torpedoboat scare a few decades before. There were those who didn't, of course, but more of them seem to have been in the Admiralty than the fleet. For example, IIRC it was the Admiratly that insisted on maintaining the "Live Bait Squadron" in the face of predictions of doom from the fleet. When that squadron lost 3 ACs in 1 day in September, even Admiralty opinion changed.
Indeed, the fleet commanders on both sides were, for most of the war, continually complaining that they had insufficient DDs to protect the fleet, let alone do anything else. This was the RN's main reason for opposing convoys so long and why the Germans never exploited Zeebrugge until they'd effectively mothballed the HSF, when it was too late. And then there were the countless panics due to bogus submarine "sightings" all through the war during operations.
By 1908, there were already large ocean going destroyers of 900 ton displacement in existence.
I would have thought that the increase in size was largely a result of a requirement for destroyers to accompany the large warships on forays overseas and counter enemy torpedo boats and coastal destroyers, rather than to provide an ASW screen.
Certainly that played a role in DD growth, but I think that was a later development, more a by-product of other things than a end in itself, at least initially.
When the TBDs were built, all destroyers and TBs were "coastal". The machinery of that day was simply too inefficient to give small ships any sort of range. The "high seas" type were big enough to get out of sight of land, cross the Channel, the Med, or the Sea of Japan, but they couldn't cross oceans. And they couldn't even go long distances along the coast without frequent refueling at sea, which wasn't possible in combat zones. So until that situation changed, DDs were incapable of accompanying the fleet in the sense meant in WW1 and later. As such, there was no reason to build them any bigger than they were already, especially because large size was still seen as a disadvantage in their primary role of night attacks.
By the time small machinery evolved to allow DDs to accompany the fleet (if sufficiently large and seaworthy), other things had happened. The earliest torpedoes had been BB weapons because fleets contained little else and their effective range and speed were on a par with BB guns and speed. But BB guns had soon left torpedoes far behind, which for a couple decades relegated them to night attacks by TBs, which in turn were countered by nets and anti-TB guns. Eventually, torpedoes achieved a range and speed to hit moving targets in a fleet action while staying out of reach of 3-12pdrs. But because guns still far out-ranged torpedoes, it was unlikely that BBs would fire them even though they still carried them. Thus, a niche was created for a true "sea-going torpedoboat", which the developments in machinery made it possible to fill.
When considering the anti torpedo boat weapons mounted on large warships, these only equate to 3pdr and 12 pdr guns (about the size of a WW2 anti tank gun) and main armament had difficulty in depressing sufficiently to fire at TB’s. The potential effectiveness of such guns needs to be considered when considering the short range of torpedoes.
In the era of the small anti-TB gun, torpedoes had a range of less than 1000m. Most WW2 tank shots were fired at ranges less than 1000m as well, and they certainly hit each other frequently enough, despite using sights no different than the anti-TB guns. And even the smallest TB was a rather larger target than a WW2 tank.
