models : the details
The details of the ECMWF (and other) model _methods_ are public, published in the scientific literature. The code is a different matter: only a handful of the _models_ are publically available, the best being those such as GFDL, etc. from the states; you can review those.
In science the best checks against error are independent _implementations_ of the experiments (in this case, weather codes), not just reviews of one piece of apparatus (eg one program). Rather than rely on code reviews (those those happen) we re-implement the methods from the papers, using different test cases. (The fixation on Phil Jone's code, etc. for example is a canard: other people have got the same results independently with their own data analyses)
There are different attitudes in met services about releasing code: in the US its mostly considered that "public tax paid for this, it should be free to the public". In the Uk, the attitude is Thatcherite : "the Met Office should operate as a business and sell what it can to keep its tax draw low", and so treats its code as prioprietary. This is why sites like metcheck.com use US models like the GFDL.
Personally I think it would be best if the codes were freely available to the public, but this has problems. Its one thing for NCAR or NOAA in the US to release their code : they have one funding source and no competition ; ECMWF is funded by 35 or so countries, who pay in to get access to the best weather forecasts and the code. If the code was freely available, why would Russia pay its fees ?
Also, be careful not to get _weather_ and _climate_ models confused. Give me a fair coin and I cannot predict heads vs tails (weather), but I can predict the _average_ over 1000 tosses ( climate). Seasonal forecasts are an intermediate case: while I cannot predict the state of the _atmosphere_ more than 2 weeks in advance, I can predict the _ocean_, which is predictable over years, because it changes much more slowly. So it "biases" the weather. I may not be able to predict that April 10 is sunny/wet, but could predict the percentage of wet days in April, by running an ensemble of weather models based on the expected _ocean_ conditions. This is the cutting edge of research, and depends a lot on increasing measurements of ocean conditions that we are now getting.
Modern models, by the way, use resolutions of 1-2 km for weather prediction regionally (eg the UK), 10 km globally, so we do resolve cities. Climate models have coaser resolutions, but we do test them with very high resolutions (extremely expensive computationally) to check for exactly the issues you describe (not resolving cities), and its not a problem; minimal gains appear above ~25km resolution for climate models.