He then uses an example of the British West Indies collapsing as a result because they suddenly had to compete with Cuba etc, which still allowed slaves (and thus could produce sugar much more cheaply).

Competition between Cuban and British sugar is a lot more complex than this, so if that's what he really thinks that is very disappointing.

The heyday of West Indian sugar and sugar products were the 17th and early 18th centuries. By the end of the 18th century, the French colony of Saint Domingue (today's Haiti) overtook them until it underwent a Revolution in the 1790s, which destroyed its plantation economy. Cuba became the major sugar exporter of the Americas in the 1790s and up through much of the 19th century.

However, foreign sugar was already out competing West Indian sugar in the 18th century! Molasses Acts were passed in the 18th century, when the slave trade and slavery itself were still in full swing throughout the British colonial empire. The text of the Molasses Act of 1733 makes explicit mention of sugar grown in the the dominions belonging to the 'King of Spain', as seen here.

In an 1818 letter, John Adams wrote

Now Sir, will you be pleased to read Judge Minots History Vol. 2d. from page 137 to page 140 ending with these Words “But the Strongest Apprehensions arose from the publication of the orders for the Strict Execution of the Molasses Act, which is Said to have Caused a greater Alarm in the Country, than the taking of Fort William Henry did in the Year 1757.” This I fully believe and certainly know to be true; for I was an Eye and Ear Witness of both these Alarms. Witts may laugh at our fondness for Molasses and We ought all to join in the laugh with as much good humour as General Lincoln did. General Washington however always asserted and proved that Virginians loved Molasses as well as New Englandmen did. I know not why We Should blush to confess that Molasses was an essential Ingredient in American independence. Many great Events have proceeded from much Smaller Causes.

John Adams is trying to argue that British attempts to implement protectionist policies for West Indian sugar, which hurt existing trade with cheaper producers who were subjects of other monarchies, helped strengthen discontent in the colonies in the lead up to the American Revolution.

So, it is patently not an issue of 'but they can still use slaves!' That was just the newest excuse, although it likely also affected the price of West Indian sugar and thus hurt it even more.

What were the causes then?

Eric Williams argues in Capitalism and Slavery that the essential issue was that most of Great Britain's most important sugar islands were small in extension and that by the late 18th century the best land had been cultivated and exhausted. He points to declining yields per acre throughout the British Caribbean.

Sugar plantations tended to practice intensive agriculture, investing little in fertilizing the soil and just trying to get as large a crop as possible by depending on the soil's natural fertility. When that soil was exhausted, they'd just buy more and exhaust that. Since the cultivatable land was pretty limited on tiny islands like Barbados, they were exhausted pretty quickly. Once they were worked to the point of a crisis in low fertility, few planters were interested in putting in the work of systematically fertilizing them, leaving them alone for a time so that the chemistry of the soil could recover, etc.

Sugar plantations also require a significant amount of lumber and sticks to keep the fires of their purging houses going. Sugar canes must be crushed to extract their juice. The juice is then boiled (which requires a good deal of firewood). Once water is extracted, much of what is left is either molasses or crystallized sucrose.

While the British empire preferred to prevent their colonies from processing and refining the sugar to the point of being ready for consumption (again, see Eric Williams as well as Manuel Moreno Fraginal's The Sugarmill), they did require some amount of refinement to take place so that the sugar would be in a 'stable' state which would weather being shipped back to England.

As such, firewood is key and while dried out sugar cane husks (bagazo, in Spanish) started to get used here and there, these worked best as a supplement to, not a replacement for, firewood. Having limited access to firewood or the need to pay high prices for good firewood would also drive up the price.

As I mentioned above, English policy of forcing planters to send sugar back to English refineries for final processing before permitting it to be exported abroad was another major issue. Sure, this brought jobs to English workers at the refineries and kept the British West Indies dependent on the mother metropole, but it also created an unnecessary extra step of embarkation, processing, reembarkation, etc., which competing sugars didn't necessarily have to go through. Cuban sugar, for example, went through the whole process entirely on the island. Some of what they'd sell was lower quality sugars for foreign refinement, but that was a choice, not an obligation.

In the case of Cuba, the island is absolutely massive with a large amount of fertile soil and virgin forests. Cuban planters could expand near infinitely while Barbados and other British isles often went through their golden age within a few decades, by which time the best soil had been used and exhausted. Haiti/Saint Domingue was also notably larger than most British sugar islands, which partially accounts for their success as well.

I don't doubt that planters tried to excuse their shortcomings by blaming their inability to import more slaves, but given these other factors I doubt that's the real reason. In short, if that's an accurate depiction of the author's beliefs, he sounds like he drank the 'West Indian planter Kool-Aid'.

Edit: As I recall, Volume 1 of The Cambridge Economic History of the United States also addresses this in part in Chapter 7 Economic and Social Development of the British West Indies, from Settlement to ca. 1850, by B. W. Higman, found here.