Ignoring the lack of high temperature alloys there was an aweful lot wrong with the German jet engines.
The major one was the decision to develop the jet engine to start with. This was forced on the Germans who had no access to very high octane fuel which limited the development of their piston engines. The jet engine appeared to be the obvious solution. It was also considered to be simpler and cheaper to make. The result was an aeroplane like the ME262. The lowest relative speed between an ME262 and a B17 would have been around 270mph. The ME262 had four cannons each with a rate of fire of 700 rounds/min. If the pilot of the ME262 opened fire at 1/2 mile that makes a total of 28 rounds fired before it passed the B17.
The axial compressor was the Achilies heel of the Jumo and other German engines. Its pressure ratio was just over 3:1 compared with a Goblin's of 3.5:1 and 4:1 for a Whittle or Rolls-Royce compressor. The main problem with an axial compressor is starting it (not the engine). When running at and below idle the compressor is far from its design point: The front of the compressor is trying to swallow too much air and the compressor is likely to stall/surge. This made starting the engine very difficult and surging was alwaysa possiblity at any speed other than its design speed. This was the cause the engine's handling problem. The Allied axial compressors being developed during the war had a blow-off valve half way down the compressor. Later on variable guide vanes and two spool compressors also came into use.
The early Allied engines used centrifical compressors because of the known technology, superchargers, and ease of handling. The Allies problem with the jet is that they did not know what to do with it. They had air superiority any way.
It is interesting to note that only one country tried to evolve an engine from the German engines and that was France. It took them years. The Russians soon gave up and restarted the development of an pre-war engine. As a stop-gap they built Rolls-Royce Nenes and Derwents.
I think you will find that many Meteors and Vampires are still using original engines although I have to admit my working knowledge of such things is from over 15 years ago. I would sooner trust a Nene or Goblin than an early RR Avon as used in most of the Hunters.
Rant of the day over.
JA