What would EEG recordings reveal if their resolution was better?
The temporal resolution of EEG is already considered to be very good. The problem is spatial resolution. Even the loss of very high frequency activity, like individual spikes, is really a problem of spatial rather than temporal resolution: the spiking cells are too far away, so it is only possible to detect them if they are very synchronized (in which case they still show up as lower frequency signals).

Because EEG recordings are so far from the sources they are recording from, and because even structures deep in the brain can contribute to surface potentials,the signals end up as complex averages over a large space. There is an entire field called EEG source modeling that tries to use signal processing methods to improve this spatial resolution.

However, you don't really have to ask this as a hypothetical question, since we already have methods that have better resolution than EEG. Electrocorticograms (ECoG) are the most similar, and most applicable to human studies. ECoG arrays are very similar to EEG, but they are placed directly on the dura, under the skull. They increase spatial resolution because they cut down distances dramatically by not having to record through the skull.

As you mention in your question, local field potentials (LFP) get even closer to where the measured currents are occurring, by recording from within the brain itself. The spatial resolution of LFPs themselves can also be improved by recording from multiple electrodes and computing a current source density. This also helps mitigate the effects of volume conduction.

fMRI is a completely separate issue. Often, EEG and fMRI results can act in conjunction, using EEG to determine time and fMRI to determine space (note that although there are limits to the spatial resolution of fMRI, for a human subject it is dramatically more spatially precise than EEG, especially for any structures not at the very surface of the brain, including any of the sulci of neocortex). fMRI spatial resolution can be improved by using stronger magnets, but eventually there are technological (and expense) limitations, but as you say the temporal resolution is fixed, and even the spatial resolution has some practical limitations.

To generally answer your title question: if the spatial resolution of EEG is improved (practically, this means either source modeling or using ECoG or LFP), one is better able to identify the brain structures that are producing the signals observed in the EEG, and able to be more confident that the signals observed are "real" rather than an artifact of filtering over space. For example, a higher-frequency traveling wave could appear as a lower-frequency signal on an EEG, or a localized, high-intensity event could appear like a generalized, low-intensity event; higher spatial resolution would allow you to distinguish between the two in both cases.