That is to say, is the deep structure supposed to be what's happening in our head when we speak a language? Or is this just to make our model of a grammar consistent?
The distinction to keep in mind is competence and performance. Chomsky is exclusively concerned with competence, ie. the knowledge of the language system. Deep structure is what defines the competence but it is not active in the process of performance (except in some very constrained contexts such as garden path sentences. Nobody assumes that we're actually generating sentences from some deep structures during the process of utterance. However, the deep structure places constraints on what can be said and what is said.
Of course, Chomsky (and most generative linguists) don't actually use the concept of deep structure any more. The earlier concepts of deep structure have moved to the notion of Universal Grammar. There are some theories like the Prague Functional Generative Description framework that postulate something akin to deep structure (they the FGD people call it tectogrammatic level) but they still would not claim this to be a part of performance.
However, your summary and the actual question are contradictory. I-Language is NOT "what's happening in our head when we speak" that's a description of performance while i-language (for Chomsky, if I understand him correctly) is still a part of competence. It's just a different way of looking at what competence is. It is contrasted with e-language to cover all the externalities like communicative and social knowledge.
Finally, it's precisely this sort of question that has led many linguists (myself included) to abandon the performance/competence distinction and postulate the usage-based model of language.
Deep structure appears to be very real. The modern research on the topic of syntax supports this notion to such a degree that the syntactic movement of constituents is basically a truism. Some fairly revealing, though obvious interpretable, data on the topic of the essentiality of deep structure exists in many fields. One of those fields of research is that of syntactic complexity.
Many experiments have demonstrated that until a syntactic constituent has found its 'original' location, the mind uses up resources 'holding on' to that constituent. Experiments will, for example, produce paradigms were an individual is presented with an apparently moved constituent early on in the sentence, and then later down the sentence place ambiguous anchor, that immediately after processing a certain word, there arises a possibility for there to be an appropriate location for the moved constituent to be located. As the sentence continues, following words disambiguate the potential 'deep location' into an incompatible location. This causes a fair amount of processing strain, and this strain is incremental as more and more 'false origins' are presented.
While this doesn't exactly prove that there is a neurological 'structure', it certainly supports the hypothesis. Almost all investigation into the topic reveal very sophisticated phenomena, going from people 'perceiving' prosodic qualities which are not there in anchoring locations, to people conceptualizing the utterance in its basic-most structure.
Deep structure seems to be so representative of our I-language that we will even commit ungrammatical utterances during communication to grammatical and deep structure-like sentences. Chomskyans, and many other linguists, really believe the deep structure to be the intensional (the 'actual' structure) of I-grammar, and for movement to be executed by the parser, which is a component that operates with the grammar, but isn't the grammar. I'm not sure, but I think that evidence for deep structure is even found in aphasiac studies.
The way that you say to make our model of grammar consistent, is I think to mean 'to make our external analysis of grammar' consistent. That is what we call 'extensional' analysis. Most modern linguists are postulating theories concerning the 'intensional' (structural) properties of language. They're saying "that's what's actually happening in the brain".
Sorry for going on so long!! Thank for reading!