I think your first statement wraps it up well, but your second is a common misconception.
SNIP
The Allies radar in WW II was neither crude (for the time) nor cobbled together. They were developmentally ahead of the curve in relationship to the Axis in every way in the field of radar from about 1936 on, despite everyone being on pretty even footing in about 1934.
For example, the German Seetakt, Freya, and Wurzburg, radars were the equal to or even slightly ahead of their operational Allied counterparts on the days of their individual introduction to operational service. They were not ahead of what the Allies had in development and prototyped. The Germans used these radars by the thousands, and made gradual improvements to them along the way.
The Allies made the CXAM/SCR-270/271 and SCR-268 (these compared well to the Seetakt and Freya at the time of their introduction) and used them by the thousands. Instead of making large improvements to these radars the efforts tended to be more minor improvements, and "lets build a new radar while we keep these models in operation". They developed a good radar, fielded it, and instead of improving it (other than required improvements to correct deficiencies) they continued its production while starting up more advanced programs in the background. Remember the SCR-270 detected the Japanese raid on Pearl Harbor on the way in with lots of warning time, but that information was not used / interpreted correctly.
By 1943 the comparative acceleration rates of the Axis and Allies in this field was pretty obviously leaning towards the Allies. The Germans were largely maxing out the ability of their existing systems, with few new systems fielded in any number, and the Allies were introducing whole new systems with new technologies. An example of this: what would become the SCR-584 started development roughly one year after the Wurzburg development started, and was operational by early 1943, 3 years after the first Wurzburg. It took longer to develop because it was doing new things in new ways. This effort paid off handsomely, as the -584 was far superior to the improved Wurzburgs of 1943. The -584 was trailer mounted and could be quickly set up (in minutes) in a new location. The -584, despite having a more compact antenna, had greater tracking accuracy. The -584 had search modes the Wurzburg did not have. The -584 had the ability to automatically track a target, were the Wurzburg was an operator intensive aided / manual track. The -584, with the M9, resulted in greater gunfire accuracy for larger numbers of guns.
The SCR-584 was in its spring with lots of headroom to grow at the same time the Wurzburg (and similar German systems) was in its fall. While aspects of the Wurzburg designs were used in very limited numbers after WW 2 the SCR-584, in various incarnations, is still in use today. The Russian built Fire Can radar (SON-9) was a copy of this radar developed from the SCR-584's given to Russia during Lend Lease and is likely still in use with several militaries around the world. It was for sure shooting down American aircraft as late as 1973.
The last time I personally had hands on an operational SCR-584 was in the early 1990's.
T!
Deliveries of Wurzburg D to the Luftwaffe started in June 1941. It gave the Luftwaffe a radar with 0.3 degree accuracy elevation/azimuth accuracy with conical scan and 25m accuracy in range with transfer to the FLAK predictor by Synchro. These figures assume a high elevation angle so that there would be no interference with the ground plane otherwise 0.5 degree and 40m. This was the finest anti aircraft gun laying radar in the world till mid/late 1943 when SCR-584 was rushed to Anzio Beachhead because the Germans were jamming the SCR-268. There was nothing like it in the allied arsenal for 2 years. Wurzburg-Riesse, with a giant 7m dish had twice the angular accuracy again and when this fixed radar as a FLAK radar was a very accurate blind fire radar to defend cities and high priority targets. Wruzburg-C (early 1941) had the conical scan but lacked the range precision and synchro transmitters and the Wurzburg-A of 1940 lacked the conical scan entirely.
SCR-268 was 2.5 times less accurate in bearing accuracy and 3.5 times less accurate in range to Wurzburg-D. That made it essentially ineffective for pure blind fire.
Around late 1943 as SCR-584 was entering service the Germans introduced the Mannheim FuMG 64 radar. As it also had a 3m dish it is often confused with Wurzburg. Output power was doubled, power drive was added, precision electronics for tracking was added and range accuracy was 10m. The second issue of this radar of early 1944 added electronic tracking and reduced ranging precision to 6m. It could track through windows once a target was found.
This radar compares very favourably with SCR-584. However the Germans had stayed on a 53cm wavelength instead of 9cm microwaves so there radar beam was wider and more vulnerable to receiving jamming energy.
In mid or late 1944 they field tested a few Rotterheim radars consisting of LMS10 magnetrons (copied from captured British CV64 but set to 8.4cm instead of 9cm to avoid interference with German passive detectors) using a Mannheim station and a Wurzburg dish. Worked quite well and was not jammed. However by late 1944 the war was going badly. The Sanitas works that made the Magnetrons was bombed and so the Germans lost all of their mass production output of magnetrons. The Hareaus Vacuum Foundry was bombed which means they lost their capability of producing permanent magnets and forcing them to modify the magnetrons for electromagnets and adding an unwelcome precision power supply.
This pretty much destroyed the deployment of microwave radar for the Germans though they managed to get up to 100 in service including on naval installations.
Below is a picture of the Marbach Egerland radar system. The flat antenna is a PPI 8.4cm microwave radar. The parabolic dish was a 4.5 m dish also on 8.4cm. The PPI radar could observe multiple targets. A pip could be placed on a target and the parabolic dish would slave to the correct direction and scan for height. Up to 7 targets could be tracked. The parabolic dish could also direct FLAK. Power output was only 16kW but 500kW and 1MW magnetrons were in the lab.
As far as Seetak goes this was the finest radar of its type till about 1943 as well and matched allied systems till the end of war.
Early versions had only 1.5kW output power, they then switched to 8kW power and shortly thereafter added what is incorrectly called lobe switching for full blind fire capability (about 0.2 degrees) it seems in 1941 on FuMo 27. Whereas the lobe switching of American Battleship and cruiser radar switched both the transmitting beam as well as the receiving the Germans had worked out a way of switching only the receiving dipoles in a procedure call raddle peilung (rattle direction finding) named after the rattling sound made by the mechanical switch of the receive dipoles. This reduced the beam width, hid the lobe switching pattern from jamming techniques and made the antenna smaller. The lobe pattern on US Battleships was so wide (15 degrees) it could confuse nearby targets.
The whole Scharnhorst sinking reinforces the myths. The Scharnhorst was observing radar silence and passive sensors had mistaken Burnets cruisers for a distant convoy though a short 30 second 'action stations' was sounded indicating the Scharnhorst had detected them probably by passive means. The short fire fight damaged Scharnhorst forward radar meaning she was no longer able to detect Duke of York in the latter encounter though again there was an action stations before DoY fired.
The German radar was 8kW at the time and was equal in performance to the British 25kW radars. The Germans had a much longer pulse of lower peak power so the range was about the same since the energy was the same. Because they used grid instead of anode modulation they could synchronise the phase of the transmitter to receiver and recover phase information for recovering resolution and accuracy. In the case of Duke of York the Type 284M radar had just been upgraded from 25kW to 125kW so the British were now outranging the Germans. The rear facing German radar could probably detect and range the British ship but at long range it probably couldn't pick up shell splash which is needed for correcting fire. At one point the Duke of York had to stop firing because it couldn't pick up shell splash as the range opened up. Scharnhorst was 50% of Germanys capital ships and was not quite the radar priority but they had planed to upgrade to 120kW power.
On the day she was sunk Tirpitz detected the Lancasters 30 minutes out at 150km
over land using her own radar. This long range indicates she had been upgraded to 120kW. She also had a Hoehtenweil PPI radar added by then and a FuMo 81 Berlin 9cm microwave radar. If Tirpitz had of ended up in one of those Iowa/Missouri fights the Tirpitz would not have been outperformed in the radar area. (To be fair H class battleship is same Generation as Iowa)
The failure to intercept the Lancasters is now thought due to treachery as the commander of the German Wassermann radar station which had detected the Lancasters was a traitor along with two colleagues they misreported the Lancasters heading. The men were latter extracted by Norwegian resistance and had planned to shoot their own officers and surrender in case of an allied invasion thus proving they had turned.
Me 109's were scrambled but were delayed further by 5 minutes due to a Ju 52 transport landing and a few minutes by being dispersed at the wrong end of the runway due to headwinds. They missed the Lancasters by 2 minutes.
