It was not by sheer chance that Groves et.al. showed up at Alamogordo with plutonium, in a sphere of exactly the right size, containing a carefully-constructed initiator, and surrounded by explosive blocks of exactly the right shape, with their detonators connected to timing circuitry specifically developed to have the necessary accuracy, and all components thoroughly tested. If it had not detonated, that would have been on account of a chance event.

And long before that could have been done, they had to do a lot of research and engineering, much of which was dead ends. US could afford to fund all that, including the dead ends. OPs argument above is that Germany couldn't compete, because they didn't have such resources, and had to pick one direction and stick to it. My point is that it's true in general, but there was a chance that Germans would have picked the right one from the get go, and managed to get similar results with a lot less effort.

Ah, I see what you mean. I think it is clear, however, from what they did do, that they were heading in the wrong direction and had some serious misconceptions that would have prevented them being lucky in that sense. There is the question of whether they even did a correct calculation of critical masses prior to hearing about the Hiroshima bomb (while being held at Farm Hall), and they apparently did not understand the relevance of prompt neutrons. I believe one of the Alsos team members said that the German scientists were lucky they did not kill themselves with their primitive experiments.