To answer this, we need to be aware of several points:
These arguments are not Kant's own
For all antinomies, Kant paraphrases historical arguments that have been made on the respective problems. These are not Kant's own arguments. Rather, Kant does not "solve" but rather "dissolve" the antinomies by offering a third way which makes the theses and antitheses only superficially contradictory.
This particular argument is Aristotelian
The argument against an infinite regress of causes is from his Metaphysics and is only paraphrased (and slightly modified, as all arguments in the antinomies...) here. Thus, the question should not be directed at Kant, but at Aristotle.
How can the argument state it that way?
The sufficiency of determination is an explicatory one: Only if a cause can completely explain the current state of affairs, it is sufficient. But if every cause in itself has a cause, then it is literally not self-sufficient, ie. it needs another cause to explain how it came to be etc.
Thus, assuming an infinite regress of causes, even the whole chain does never sufficiently explain the outcome as we will never reach an ultimate cause that itself is sufficiently explicatory. The explanatory force gets handed down infinitely, as it were.
What does Kant himself offer?
Kant, like discussed shortly in this answer of mine, with a pointer to where Kant writes about that does distinguish between infinite and indefinite. For any causal chain of an empirical event, it would be wrong to say there is an infinite chain of causes since the event and the causal chain are mere representations of reality and thus our picture of it. According to Kant, we always have to keep in mind that we cannot easily apply what and how we think about the world to metaphysical truth. Indeed, the whole book is about how we must end this in order to finally provide a solid fundament for any proper philosophy. Accordingly, he says that all we can say about any empirical causal chain is that no matter how far we try to go back, we so far have ever found another cause behind the last one. Thus, we can say there is an indefinitely long causal chain here.
The argument about the explanatory insufficiency does apply all the same though.