How can neuronal signals faithfully be reproduced by scalp electrodes?
EEG has been recorded beginning in the late 20s, by people who were fairly clueless about electricity. While the strength of the EEG is indeed weak, it is readily observable with even such comparatively primitive technology.

The first thing you should take note of is that EEG measures the aggregate field. It does not see single neurons. It sees the synchronized activity of large neural populations of neurons - tens, hundreds of thousands of neurons at the very least, and probably more on the order of tens and hundreds of millions of neurons. This aggregate activity creates activity on the order of microvolts. This activity is then relayed (via the differential amplifier technique) to a powerful amplifiers. Modern EEG amplifiers have very large input impedances, and are thus able to amplify these weak signals. However, it still only observes mass action of large groups of neurons, not e.g. single action potentials by single cells.

Next, direct EEG analysis is comparatively uninformative for many reasons, including those you mention: that the scalp, the skull, and the other intervening tissue attenuate the signal. Thus, very often, extensive preprocessing is applied, such as averaging over many recordings. However, at least two features are already visible in the waking EEG. The first is the power-law structure of the frequency spectrum (although this is reinforced by the filtering characteristics of the tissue; the tissue filters higher frequencies more than lower ones). The second is the alpha wave, which is a large activity pattern coming mostly from visual areas. However, this is a very coarse feature; it tells us little beyond if the subject has their eyes open and is attentive, or has them closed/is inattentive.

Lastly, it is indeed generally assumed that deep sources (such as subcortical areas) are not picked up by the EEG (but only by e.g. MEG).

A good introduction to the topic is the book Electroencephalography by Niedermeyer and Lopes da Silva.