I'm not sure referring to them as RADAR pickets is appropriate ... PICKETS, yes ... Take a look at the standing orders regarding HOW MANY MEN were on deck observing the skys with field glasses while a ship was on picket duty ... The REASON those Pickets were required is that WHEN RADAR FAILED to detect Bogeys at Long Range ... There would be a BACK UP Spotting screen that would see them in time for the Capital Ships to get to Battle Stations and prepare for Kamikaze attack. Radar was a reality, but NOT a reliable one, even by the end of the war.
Go look up the Sumner and Gearing class destroyers - that big antenna you see is an actual HF (height finding) radar array. By 1945, these guys are not only finding aircraft targets at range, they are talking directly to intercept aircraft and feeding them altitude data. These destroyers were some of the first built with combat information centers to coordinate interception.
http://destroyerhistory.org/sumner-gearingclass/gearingclass/No radar isn't a 100% guarantee of detection in 1945, as there were countermeasures - jamming, chaff, flying low below the radar horizon, etc. that you could take. Having a load of guys on deck with binoculars is what alerts you when a load of guys come in fast at wave top level (or if a periscope pops up a couple of miles away). This is not a problem with the radar "reliability", just the problem posed by line of sight to a target flying at very low altitude. When put in aircraft, these "unreliable" radar sets were perfectly able to pick out U-Boats at 40 miles, and smaller targets like conning towers at 15-20 miles (look up the ASV10 centimetric radar sets on B24s - used after 1943).
There is a VAST difference between the early 1940-41 sets and the late war sets. It was much less a "reliability" question than one of radar frequency and how much power you could put out. Late 1930's and early 40's stuff was very low power and hence short range. By 1945, they are picking out aircraft at 200 miles and frying birds on the antennas.
Interesting shot of the CV16 (Lexington) Radar Arrangement in 1944/45:
http://en.wikipedia.org/wiki/File:CV-16_1944_radar_arrangement_NAN3-46.jpg