I believe the question is confusing the difference between something being a natural basis for a language, versus whether something could be adequately represented by one in a mathematical / encoding sense.
It is clear that morse code can be represented adequately using a binary representation / encoding. This is not surprising, since a binary representation can be created to represent much more complex bases anyway, such as the latin alphabet (e.g. ASCII).
However, you would probably agree that this doesn't make the latin alphabet a binary one in nature. Rather, the alphabet's elementary particles (the letters) can be used to form more complex constructs (the words). So in this sense, the English language is most naturally regarded as a 26-base system (because there are 26 elementary letters-particles).
Similarly, while it is possible to represent morse code using binary encoding, one would be hard pressed to argue that this is the most natural representation for it, or that this makes it a binary 'alphabet'. It is rather intuitive to consider the 'dit', 'dah', and 'separator' as the elementary particles that combine to form more complex constructs ('morse code').
You could of course try to argue that the 'dah' is not a well-suited, natural atomic particle for the morse code language, hence you can accept the 'binary' representation of '10', '1110', '00' etc, but I would argue that this is not really a good interpretation of the nature of morse code words; it is far more intuitive to conceive of morse code letters as composed by elementary dits, dahs, and separators as the elementary particles. If you need three bits to express a 'dah', then you're using essentially using four 'atomic particles' in your chosen representation, to express what is essentially a single natural atomic particle in the morse language, which seems like a rather inefficient way of expressing the language. Remember that binary is the choice of digital signals because of the electrical nature of transistors which deal with such signals most efficiently. But there's no reason for morse code to abide by that requirement, so a more efficient representation which is more natural to morse code (and its transmission by telegraph) would presumably have been preferable to such a binary encoding among telegraphers.
As to the separate question of whether it should be possible to consider the dit and dah itself as a sufficient number of elementary particles to fully express the morse code language, the answer to this is no (and hence it is not a naturally binary 'alphabet'). You can confirm this by attempting to express morse code in the absence of spaces. The reason telegraphers could rely on only dits and dahs is because they also had the added element of time, which could simulate word separation. When these need to be written down on paper as symbols, however, the word separator needs to be made explicit, thus making morse code a ternary system.
Another way to see why time in itself is not a trivial aspect, but actually adds information is to treat it as if it's a separate signal transmitted in parallel with your dits and dahs. You now have two signals to consider at each time point, one in the dit-dah dimension which tells you whether you're dealing with a dit or a dah at that timepoint, and one in the time dimension, which tells you if you're dealing with a dit-dah particle, or not (i.e. a separator). Since these are two independent binary signals, your receiver at the other end would have to process a 4-bit signal. However, this can be encoded more efficiently as a 3-bit signal, since when the time series has a separator, the ditdah series is ignored. Thus the most efficient and natural representation for morse code is a ternary one.
PS. I forgot about the distinction between a 'letter separator' and a 'word separator', but the above arguments still apply. You could make the case for a quaternary base instead of a ternary one incorporating a bespoke 'word separator' particle, or accept the ternary one and accept the inefficiency that comes with always having to represent a 'word separator' using a slightly less efficient representation using a 'diphthong'.