US skippers were still afraid of DF wrt their radars as I recall. I think they'd turn the radar on for a while every hour, then turn it off. Later in the war you read of surfaced "high periscope" patrolling, too. Earlier in the war, submerged during daylight was pretty common for fleet boats, though (they had a 2 day endurance submerged at 2 knots or so).
"Keep them down" is indeed the crux of ww2 ASW doctrine done right. The Japanese actually had this notion (read the post-war interviews in the strategic bombing survey), but their execution was less creditable. The Allies not only had and used this doctrine, they also had sigint and code breaking. The sigint alone would have utterly doomed the u-boats. It's not just sinking subs, but simply putting ASW assets where the u-boats are forces them to stay down. Combined with putting the convoys where the hunter killers (and u-boats) are NOT, and you've crippled them.
For all the amazing tonnage sunk by u-boats (11-12 million tons, right?), it was still only a few ships sunk per sub lost (~10,000 tons per boat lost). A merchant ship, banged out in a few days or a week, crewed by at most a couple dozen guys is a good trade (particularly with a good % of the crews rescued) for a far harder to produce sub, and what, ~4 dozen highly trained men?
US subs sank a fraction of the tonnage, <5 million tons, but it was ~100,000 tons per boat lost.
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