Is there a reliable objective study that addresses this issue?
No. There are several dimensions that would have to be addressed. I can hint at what they are, with the intention of exposing some of the inherent conflicts of such an inquiry.
First, there is no doubt that various laws and court cases that were established in the 1950s and 1960s ended institutional racism. Johnson was instrumental in some of these, but they were not necessarily 'Great Society'. The Civil Rights Act of 1964, in particular, was not an attempt at an economic program, it was focused on eliminating state laws that institutionalized discrimination. 'Great Society' acted pro-actively, often but not always in economic terms.
Head Start is one educational program that most view as an unqualified success. Over the span of it's existence, it has consistently helped the children that were enrolled in the program.
The Higher Education Act of 1965 vastly expanded federal funding of higher education, including the provision of low interest student loans. The value of this is probably described as a bell curve, with positive and negative aspects evenly distributed around the mean. On the one hand, many people who deserved access to higher education were able to act on it. On the other hand, many of those who enrolled were viewed as not having met the same qualification criteria as non-minority students, and felt ostracized. Retrospectives on this aspect suggest this caused real harm.
It may also have created more systemic hazards. As the Federal Government supplies more and more of the money to higher education, it also politicizes campus policies, and has created severe distortions that in some cases impinge on student freedom of speech and due process. It also appears to be suffering a problem similar to crop subsidies, more students are graduating with degrees than can find jobs that require college educations. This can negatively affect minorities that don't have college or even high school educations, as they are crowded out by non-minorities with more complete attainment.
Larger Economic Framework
The industrial revolution sparked a collection of redistributionist philosophies, Marxism being one of them. While social systems are always redistributionist to some degree, the doctrine of economic central planning didn't really take hold until Stalin took control of the Soviet Union. Just because the communists took over in 1918 doesn't mean that the central planning apparatus suddenly materialized out of nowhere, this took time to implement. The experience with Fascist Italy and Nazi Germany also appeared in the 1930s, particularly after Germany had been through the hyperinflation of the 1920s. Therefore, at the point in time that MLK was proposing a restructuring of the US economy, 'the facts weren't in'. While the German and Italian economies had been released from centralized authority, little was really understood about how the Soviet experiment was running. It would be another decade or two for the bankruptcy of that order to be certain.
In the meantime, in the period from 1945 to 1972, the US 'owned the world'. Europe and Japan were rebuilding from nearly total destruction. The US population exploded from about 135 million to 200 million between 1945 and 1963. This triggered massive construction of housing, freeways, aircraft, telecommunications infrastructure, and consumer goods. With the arrival of television, those that were locked out of this economic bounty were seeing it day to day, one can review any number of the classic 1960s sitcoms to imagine how that was being received by people living in tenements.
The idea that someone halfway through their life in the 1960s now had the 'opportunity' to get better employment, was, overall, pretty useless. Without education or experience someone at age 35 was not going to shift from warehousing to doctor or attorney. This would take time, for many, they had been waiting long enough.
Given the economic power of the US at the time, the idea that it should be shared had a lot of political support. Thus it was easy to enact rafts of legislation for Medicare, Aid to Families with Dependent Children, Food Stamps, student loans, etc. Many of these didn't go all that far at the time, such programs were expanded later, both in terms of services provided and as to who was served. This has created a massively larger government, and is coincident with deficit spending and rapidly rising government debt.
The legislation that was enacted in the 1960s would have been 'affordable' if it had remained in it's original scale. The two problems were that the scale expanded, and that people became dependent - not merely as recipients but as government workers employed in the vast bureaucracy. Many of these workers are minorities - people that wouldn't have had such jobs had the agencies didn't exist. Needless to say, as government employees they are better off than they might be sweeping streets, however in this role they do not face market forces, and therefore have little incentive to adapt to changing circumstances. What we are finding now is that attempts to simply downsize government and reign-in public sector benefits is a 'third rail', touch it and you're dead. As this continues, the government continues to draw a disproportionate slice of the economic pie, leaving everyone worse off.