What is it that people mean by term "energy independence"? The resemblance to the term 'political independence' is relevant, and the second term, with all its rich history of personal sacrifice probably warms the first by near proximity. But these terms are very different, or must be, since energy and politics are themselves so different. Energy is something consumed (or at least used; the low entropic state is used and lost, so perhaps we can say consumed). Politics, on the other hand, is something that people do. Independence has very different senses in relation to these terms. In relation to politics, it suggests action free from outside influence. A citizen can be said to have political independence when they are free to choose any course of political action available; a population may be said to be politically independent when its government, however arrived at, is able to act freely without taking orders or being in any way constrained by another power. Energy independence cannot be analysed in the same way. A private consumer can never be independent of the producer unless he or she is also the producer and is independent. A household that grew food, wood, generated off-grid electricity, and perhaps also made biodiesel or used horses pastured on their own farm, might be said to be energy independent, but the criteria are very strict and it is not absolutely clear that the degree of independence is complete, since the equipment bought to assist in the growing, generating and converting of the energy would be the product of energy consumed in other places.
A population, say that of the United Kingdom, might be said to 'energy independent' if it never imported energy from producers outside its jurisdiction, though the question might be raised with regard to the energy rendered as the complexity of imported goods. But the UK is not the importer, individual companies are; and the consumer engages with them simply as the most proximal element in the production chain. From the consumer's point of view the origin of the energy has no significance, except that one source may be cheaper and the other more expensive, and there is no evidence to suggest that imported energy is necessarily dearer than indigenous energy; it will vary from place to place and time to time. To put the point demotically: the consumer doesn't care where the energy comes from; they care about price, quality and reliability, and there is no intrinsic benefit to indigenously produced energy on any of these metrics. Indigenous energy could well be more expensive, poorer in quality, and less reliable than imported energy. British wine is all of these things. Or, compare electrons from British wind power and from the interconnector with France.
From the consumer's point of view, energy independence is an empty concept, and its use is purely rhetorical; indeed it is suasive marketing, probably intended to cover disadvantages, such as high cost.
The only benefit arising from indigenous energy is that it introduces further competition into the market. External producers will already be competing, but if there are also internal producers then there will be more competition. This is good, but additional competition can also be found externally. It isn't an intrinsic and exclusive property of indigenous energy.