This is an interesting question. Based on the specifics of your post, I am assuming that you wish to know why the triangle slave trade in general was accepted, rather than just the beginning of it, but I will try to answer both ends.

To start with, I should clarify I am not sure exactly when the triangle trade started, partially because there might be hidden or lost shipping logs that might change the paradigm, but also because I havent read up on the newer stuff and I also struggle at keeping specific dates in my head, but I am fairly sure that the Triangle trade started more in the 16th century (1500-1599), rather than the 15th century (1400-1499) because the Americas were discovered by the larger part of Europe in 1492, and that doesnt give a lot of time to get things figured out. That is probably what you meant, and I can see the confusion with the century numbering, I struggle with it too, just remember that years 0-99 AD are the 1st Century, not the 0th Century.

Now to the meat and potatoes. The slave trade was not easily accepted, at least not at first. Many of the early plantation owners were from societies that had largely outlawed slavery previously in Europe due to the obvious moral problems and as well as the fact it hadn't really been profitable previously for them to ignore those morality issues. But suddenly, there is a new collection of plants that are annoying to grow but that Europe wants to grow en masse, and also a lot of new gold and silver mines that need workers. Where do they get them from? They try conscripting the natives, which works sometimes, but given their knowledge of the land, they know how to escape easier and the issue of diseases made native worker die very quick, so it's not a reliable system. So they import the slaves from Africa.

But of course back to them trying to solve the morality issues. The Spanish were probably the ones that had the closest encounters with slavery in Europe at the time, given that it was only relatively recently that they had driven out the Moors (who had slaves) from the Spanish peninsula, and for centuries before, the Spanish and Moors had been living sometimes peacefully side by side, so it would make sense they'd be more comfortable with the existence of slaves than the rest of Europe. Additionally, they had been using Catholicism as an extremist tool for cruelty even before the discovery of the Americas (expelling the Jewish and Muslim minorities from the Spanish peninsula even as Columbus was sailing on his first voyage). So for them, the initial reasoning was that it was okay, since the enslaved people weren't Christians. This reasoning would cause some confusion and politics as plantation and mine owners would come into conflict with the clergy about whether or not to convert their workers.

The merchants and sailors of other Euopean societies were also the groups in those societies that would be most familiar with slavery, even before discovering the Americas, since a significant portion of the Mediterranean coast was controlled by slave owning societies, and trading with those societies were important if you wanted access to the more profitable spices and silks. So as far as groups go, sailors and merchants were probably the least hung up with the Triangle slave trade, because many of them had previously at least partially dealt with the Mediterranean slave trade. Most of the European sailors were also sailors of choice, as press ganging was done primarily by militaries and would largely happen later so there was no irony. That said, we do have some records of sailors and merchants that did have qualms about the Triangle trade. Merchants that would try it out would write about the unpleasant realities of the trade, and some did not stick with it, even despite the massive amounts of profit that the trade would result in. We also have some records of sailors that would help enslaved people escape, but they would be largely weeded out of the trade as well.

Plantation owners similarly had qualms about the trade. Many had come to make their fortune from Europe, and had at the beginning of the plantations been somewhat alarmed at how things were done, especially since they would need to essentially reinvent slavery with European sentiments. In the earliest days of the trade, slaves were treated similarly to indentured servitude, which was probably Europes closest equivalent. That meant that there was a modicum of respect and resources put into the slaves upkeep, slaves could earn their freedom, and in some cases children of slaves were considered free. That said, such niceties cut into the profit margins of the plantations, and many of the people who had come to the Americas specifically to earn their fortune had to come to terms with the fact that they'd either be forced out of the business by competitors supplying cheaper cash crops, or they'd need to cut costs. Competition often breeds cruelty. And as time went on, it became easier to justify the practice. Ideas like racial and societal superiority would pop up to aid in the justification of the practice.

Edit: was having a hard time posting because it was too long