Truth be told, this goes a bit beyond my area of expertise because I focus chiefly on British colonies and their successor states, not the New World system of slavery in general. Whilst I can give you a comprehensive answer to your main question, I'm afraid I can't go into the same kind of detail I would if we were talking about, say, Jamaica if you have any follow-ups, though I'd certainly try to offer what I can.

It's worth mentioning to begin with that the answer to your question partly depends on what you mean by a 'black population'. Most people in the United States would, at first glance, probably say that the Dominican Republic does indeed have a very large black population; many people in parts of the Caribbean would not share that assessment. This is partly because the United States has not historically had a very strong concept of mixed ancestry; the experience of slavery, particularly its cultural and legal construction whereby slavery was equated with blackness and was inherited from one's mother, means American racial discourse has historically tried to categorise people neatly as either 'white' or 'black'. This is still true in many ways today: if you want to identify as mixed race on the US Census, there is no option for mixed heritage without writing it in yourself.

The Dominican Republic actually has a relatively small black population; smaller than the United States', and white people outnumber black people. Around three-quarters of the Dominican Republic's inhabitants are in fact regarded as mixed race, most having heritage that is both white and black - contrast that to Haiti where the crushing majority of the population is considered to be black and only a small but prominent minority of mixed ancestry. In the Dominican Republic, there has historically been a great reluctance on the part of mixed race individuals to embrace black or African identity unlike in most of the rest of the Caribbean; indeed in 20th Century discourse on the nation in the United States, you actually find some people making the claim that the Dominican Republic is an overwhelmingly white nation opposed to domination by 'black Haiti'.

As to the origins of the modern-day state's racial demography, what would one day become the Dominican Republic was, in fact, a slave society for most of its history. In the early 16th Century, it had been a centre for sugar production and saw the importation of a huge number of African slaves to work on its sugar estates. Unlike Haiti however, this was a short lived period of sugar dominance in the colony's economy - from the mid 16th Century onwards, the sugar industry goes into dramatic decline. In the 17th Century, the plantation system is essentially driven into meltdown by a series of economic and political crises, as well as natural disaster. This results in a widespread emancipation of ex-slaves as estates collapse into ruin and the racial hierarchy begins to disintegrate somewhat, creating a large free black population that ultimately ends up integrating with the white population. Throughout the colony's history, it struggled with slave revolts and desertions that drained considerable resources and caused significant damage - the prevalence of that kind of resistance seems to have encouraged the white elite to look on free blacks, who waited peacefully for emancipation, as more 'civilised' and dignified than the majority of their enslaved counterparts who were seen as a constant thread to the colony's security. In marrying into white families and having mixed race children, many free black people may have seen opportunities for their own children to escape the stigma of racial prejudice.

It's worth noting that this in turn perhaps helps to explain the racial dynamics of the modern-day Republic; we see that in elite discourse, 'black' becomes increasingly synonymous with 'slave' rather than with skin colour as the free black population and the white population become increasingly integrated with one another. It's a curious process that simultaneously destroys and reinforces the idea of black inferiority; free people of all skin colours increasingly define blackness as a condition of degraded servitude in slavery, and associate freedom with whiteness. Whilst that happens in the United States also, in the United States it works by preventing black people from placing themselves in wider society. In the future Dominican Republic, population constraints mean we see the opposite happen: free black people are able to somewhat escape the burden of racial slavery, rather than be defined by it. Essentially, the physical condition of being dark skinned becomes partially divorced from the social condition of being black, which becomes increasingly reserved only for slaves.

Slavery actually continues until the 19th Century when it is finally abolished - in 1800, there were still some 40,000 slaves in the future Dominican Republic. But the conditions created by the colony's unique experience of slavery obviously meant that this was a society where integration was a much greater possibility for ex-slaves post-emancipation. In the case of more recent history, it's also worth pointing out that there is also significant economic migration from Haiti to the Dominican Republic; around 500,000 largely black Haitians were thought to be living and working in the Dominican Republic in 1994, out of an official population of about 8million people (see Corto et. al. in Latin American Perspectives 22, no. 3, pp94 - 140).

So you're right on the mark when you say that the argument the Dominican Republic was never a slave society is essentially wrong: it very much was a slave society, just one that evolved along very different lines to Haiti, and where slavery was smaller scale for most of its history. Slavery still largely explains the modern-day demographics of the Dominican Republic, though they are complicated by the unique relationship Dominicans have with the concept of 'blackness'; Dominican racial discourse is filled with contradictions and curiosities. On the one hand, many Dominican ex-slaves and their descendants were able to escape racial prejudice and integrate into wider society much more effectively than in most contexts; on the other, this was achieved by creating a culture that degraded black and African identity. It was an escape from one system of racial stratification into another.

There's a very good treatment of this topic by Silvio Torres-Saillant in a journal article called "The Tribulations of Blackness: Stages in Dominican Racial Identity" (Latin American Perspectives 25, no. 3 (1998): 126 - 146) if you have access to it, which really traces in quite fascinating detail how this racial ideology emerges.